Numerical Dissipation and Stability Analyses of a Highly Efficient Numerical Approach Proposed for Predicting the Blast Loads from Large TNT-Equivalent Explosives on Building Structures

Abstract

An efficient CE/SE-based numerical method implemented in LS-DYNA R7.0 has been developed for predicting far-field blast loads from large TNT-equivalent explosives on structures, offering validated accuracy for engineering risk assessment. However, the stability and numerical dissipation characteristics induced by its non-physical parameters (α, β, CFL) remain unquantified, limiting optimal application. To resolve this, we analytically derived an explicit numerical dissipation term coefficient aβΔx(w₊ − w₋)(1 − CFL²)/4 from a simplified 1D continuity equation and established stability criteria via von Neumann analysis. Benchmark simulations of 1 kg TNT free-air bursts demonstrate that increasing α from 0.1 to 5.0 reduces peak overpressure by 13.0%, while β rising from 0.5 to 1.0 decreases it by 1.89%, and elevating CFL from 0.05 to 0.50 increases overpressure by 1.92%. Critically, stability requires α ≥ 0, 0 ≤ β ≤ 1, and 0 ≤ CFL ≤ 1. These first theoretical guidelines for non-physical parameter selection enhance the method’s prediction accuracy and computational efficiency.

1. Introduction

Accidental explosions involving hazardous chemicals frequently cause structural damage across significantly larger distances than typical terrorist blasts [1,2]. The catastrophic 2015 Tianjin Port ammonium nitrate explosion, equivalent to 450 tons of TNT, which severely damaged buildings within a 5 km radius, exemplifies this extensive threat [3]. Consequently, accurate prediction of far-field blast loads is critical for structural risk assessment but remains a significant challenge [4,5]. Existing prediction methods face substantial limitations: empirical approaches [6,7,8] are often overly simplistic; experimental methods are prohibitively costly and hazardous; conventional numerical simulations demand excessive computational resources for large-scale explosions due to the need to model extensive air domains over long durations [9,10].

A novel and efficient computational fluid dynamics (CFD) method was developed to overcome these limitations, utilizing the space–time Conservation Element/Solution Element (CE/SE) scheme [11]. Unlike conventional finite volume or finite element methods that discretize differential forms of governing equations—struggling with non-differentiable shock discontinuities—the CE/SE scheme enforces integral conservation laws directly within unified spacetime elements. This approach intrinsically captures strong discontinuities (e.g., detonation waves) without artificial dissipation [12,13]. Implemented in the LS-DYNA^® CESE solver since R7.0, the scheme integrates equations in strict conservation form. Critically, by unifying spatial and temporal discretization (avoiding operator splitting techniques used in traditional CFD), CE/SE inherently suppresses numerical oscillations at blast wavefronts [14,15,16], enabling robust simulations of far-field blast propagation. The above-mentioned developed method constructs a generalized blast wave by specifying initial and boundary conditions for disturbed air, leveraging validated empirical pressure data. Air particle velocity is calculated using an explicit formula derived from inviscid gas equations, while density is determined via a simplified path line method. The approach has proven highly computationally efficient and sufficiently accurate for engineering applications [14]. It demonstrated superior performance in predicting far-field loads from large TNT-equivalent explosions compared to alternative methods like MMALE [17,18,19,20], mapping [21,22], and LBE [23]. Validation against UFC empirical data [8] and experimental results showed maximum relative errors below 10%, achieving accuracy comparable to the LBE method and superior to the mapping method [14]. Furthermore, this method has been successfully applied to calculate blast loads on gabled frames from large explosions, enabling the establishment of an empirical blast load model [24].

The proposed method for predicting far-field blast loads employs the CE/SE scheme to solve the Euler governing equations for an inviscid compressible fluid. This CE/SE scheme introduces specific non-physical parameters (α, β) to ensure numerical stability during the solving process. However, the introduction of these parameters inevitably induces numerical dissipation (or numerical damping), which can reduce solution accuracy by artificially increasing the diffusivity of the simulated medium compared to the true physical system. Numerical dissipation, while potentially detrimental to accuracy, is often necessary; calculations with insufficient damping, such as high-order finite-difference schemes, can fail for problems involving blast shock waves [25]. The primary challenge in selecting the parameters α and β lies in balancing the need for numerical stability against the undesirable introduction of excessive numerical dissipation. Therefore, a detailed analysis of both the stability conditions and the degree of numerical dissipation inherent to the specific CE/SE scheme implemented in LS-DYNA is essential to guide optimal parameter selection for maximizing prediction accuracy. Although the stability and dissipation properties of various CE/SE schemes have been investigated [26,27], these critical characteristics remain unquantified for the particular formulation embedded within LS-DYNA.

Stability constitutes a fundamental property for any robust numerical method predicting blast loads. The stability of this CE/SE scheme depends critically on the non-physical parameters (α, β), the timestep, and the mesh size, governed by constraints like the Courant–Friedrichs–Lewy (CFL) number. Instability renders simulation results meaningless, while excessive numerical dissipation degrades their accuracy. Research in other CFD contexts underscores the significant influence of stability parameters like the CFL number on solution fidelity; for instance, elevated CFL values can artificially attenuate pressure oscillations in simulations of ethanol-fueled engine knock, obscuring critical phenomena [28]. Von Neumann analysis represents a well-established method for deriving stability conditions for CFD schemes solving the Euler equations [29,30], paralleling techniques used to analyze numerical damping and stability in structural dynamics algorithms (e.g., spectral radius criterion [31], amplification matrix method [32], equivalent damping ratio [33]). Guo et al. [34] proposed an improved single-step method of explicit displacement and velocity as numerical solving scheme for structural dynamic problems. Their analysis indicated that properties of numerical damping and stability varied with different non-physical parameters. Explicit stability criteria are vital for guiding parameter selection to prevent computational failure or impractical resource demands, as highlighted by efforts to model superheater flows without resorting to impractically small timesteps dictated by strict CFL conditions [35]. Roe [36] analyzed stability of 4 types of CFD schemes, which were the Jameson-Schmidt-Turkel method, the optimized finite-difference method, the discontinuous Galerkin method, and the active flux method. The analysis work provided conditions of stability such as the maximum Courant number or influence of frequency band on numerical stability. Furthermore, it is important to note that stability conditions can vary depending on the problem characteristics; studies on time integration methods, for example, have shown conditional stability depending on structural stiffness behavior [37,38].

This study addresses the critical gap by conducting the first detailed analysis of both the numerical dissipation characteristics and stability conditions of the specific CE/SE scheme implemented within LS-DYNA for far-field blast load prediction. We systematically evaluate the impact of the key non-physical parameters α and β, along with the CFL number, on the degree of numerical dissipation, using the derived formulae and the benchmark case of a one-dimensional free-air burst explosion of 1 kg TNT. Furthermore, we rigorously analyze the stability of the scheme via von Neumann analysis to establish the permissible ranges of α, β, and CFL that ensure stable simulations. Explicit formulae derived from a simplified one-dimensional convection problem to quantify numerical dissipation and stability conditions are demonstrated to be entirely consistent with the numerical results. The framework developed herein provides essential guidance for selecting non-physical parameter values that maximize the accuracy of blast load predictions.

The remainder of this paper is organized as follows. Section 2 briefly outlines the previously proposed CE/SE-based numerical method within LS-DYNA. Section 3 presents the evaluation of numerical dissipation, combining simulation case studies and theoretical analysis. Section 4 details the stability analysis and derives the stability conditions. Finally, a comprehensive discussion is provided in Section 5, synthesizing the key findings, comparing them with existing studies, and outlining future research directions. Finally, concluding remarks are provided in Section 6.

While CE/SE schemes have been extensively studied for blast simulation, critical gaps persist for practical implementation in commercial solvers. Specifically, the LS-DYNA CESE solver introduces unique non-physical parameters (α, β, CFL) to ensure stability, yet their quantitative impact on numerical dissipation remains uncharacterized in the literature—a fundamental limitation for engineers seeking to optimize prediction accuracy. Previous studies either focused on generic CE/SE formulations or validation case outcomes, but none: (i) Derived the explicit numerical dissipation term inherent to LS-DYNA’s specific implementation, (ii) Established rigorous stability bounds for its parameters via von Neumann analysis. This work bridges these gaps by providing the first theoretical framework quantifying dissipation coefficient as aβΔx(w₊ − w₋)(1 − CFL²)/4 and proving stability requires α ≥ 0, 0 ≤ β ≤ 1, and 0 ≤ CFL ≤ 1. These contributions enable precise parameter selection, elevating the method from empirically validated tool to physics-grounded solution—advancing computational blast dynamics for infrastructure resilience.

2. Blast Load Prediction

2.1. Blast Wave Generation

Figure 1 provides a graphical representation of the computationally efficient numerical simulation technique developed by Zhang et al. [14] for predicting blast loads on structures. This approach employs a rapid algorithm to determine the flow field parameters (pressure p(r, t), particle velocity vector v(r, t), and density ρ(r, t)) within air disturbed by a blast shockwave, defined at distance r from the origin and time t after detonation. The simulation’s initial state (t_i) corresponds to the instant the shock front reaches a designated position R_i ahead of the target structure. Flow parameters at t_i (p(r, t_i), v(r, t_i), ρ(r, t_i)) serve as the initial conditions for the numerical model. Computational efficiency is maximized by setting t_i such that the initial blast wavefront arrives within 1–2 mesh layers anterior to the structure’s front face at simulation start (i.e., minimizing d_i in Figure 1). Increasing t_i reduces simulation duration proportionally, but the wavefront must not surpass the structure’s front face to avoid invalid initial conditions. Thus, t_i is determined via empirical arrival time t_A(r) at standoff distance R_i ensuring d_i ≥ 0. Furthermore, time-dependent boundary conditions (p(r_b, t), v(r_b, t), ρ(r_b, t)) must be specified at the inlet boundary (radius r_b from the origin). These temporal profiles are applied to the model to simulate the energy and entropy contribution from the detonation and its products. While key algorithm formulae are presented here, comprehensive methodological details are documented in the original reference [14].

The air pressure is defined by the following modified Friedlander’s equation:

$$p\left(r,t\right)=P_{\mathrm{SO}}\left(r\right)\left(1-{\displaystyle \frac{t-t_{\mathrm{A}}\left(r\right)}{t_{\mathrm{o}}\left(r\right)}}\right)e^{-\chi \left(r\right){\displaystyle \frac{t-t_{\mathrm{A}}\left(r\right)}{t_{\mathrm{o}}\left(r\right)}}}+p_0,$$

(1)

where P_SO represents the peak incident overpressure, t_A is the arrival time of the wavefront, t_o is the positive phase duration, χ is the waveform parameter, and p₀ represents atmospheric pressure. The values of these parameters can be determined from empirical tabulated data and curves [7,8,39].

The air particle velocity is defined as follows.

$$u\left(r,t\right)=\left\{\begin{array}{cc}0& \begin{array}{cc}\mathrm{if}& t<t_{\mathrm{A}}\left(r\right)\end{array}\\ {\displaystyle \frac{u_1\left(t\right){\left[r_1\left(t\right)\right]}^2{\left[p_1\left(t\right)\right]}^{1/\gamma }+{\displaystyle {\int }_r^{r_1}f\left(x\right)\mathrm{d}x}}{r^2{\left[p\left(r,t\right)\right]}^{1/\gamma }}}& \begin{array}{cc}\mathrm{if}& t\ge t_{\mathrm{A}}\left(r\right)\end{array}\end{array}\right.$$

(2)

Here, u₁ is the peak value of the particle velocity profile, r₁ is the distance of the blast wavefront from the blast origin, and p₁ is the peak pressure at time t, while γ is the ratio of the specific heat capacities of air at constant pressure to the specific heat capacities at constant volume, which is 1.4. The integrated function in Equation (2) is given as follows.

$$f\left(x\right)=-{\displaystyle \frac{P_{\mathrm{SO}}\left(x\right)r^2{e}^{-\chi \left(x\right)\frac{t-t_{\mathrm{A}}\left(x\right)}{t_{\mathrm{o}}\left(x\right)}}\left[t_{\mathrm{o}}\left(x\right)+\chi \left(x\right)\left(t_{\mathrm{A}}\left(x\right)+t_{\mathrm{o}}\left(x\right)-t\right)\right]}{\gamma {\left[t_{\mathrm{o}}\left(x\right)\right]}^2{\left[P_{\mathrm{SO}}\left(x\right)\left(1-\frac{t-t_{\mathrm{A}}\left(x\right)}{t_{\mathrm{o}}\left(x\right)}\right)e^{-\chi \frac{t-t_{\mathrm{A}}\left(x\right)}{t_{\mathrm{o}}\left(x\right)}}+p_0\right]}^{\left(\gamma -1\right)/\gamma }}}$$

(3)

The air density is calculated as

$$\rho \left(r,t\right)={\displaystyle \frac{{\rho }_1\left(r_{\mathrm{insec}}\left(r,t\right)\right)}{p_1{\left(r_{\mathrm{insec}}\left(r,t\right)\right)}^{1/\gamma }}}p{\left(r,t\right)}^{1/\gamma },$$

(4)

where ρ₁ is the peak density and r_insec is the distance of the intersection of the path line and the wavefront from the blast origin at time t. The intersection can be determined using the simplified method presented in the original report.

2.2. Method Implementation

The scheme applied for implementing the proposed method is illustrated by the flow chart in Figure 2. Specifically, meshes in front of the target structure should be chosen to simulate the air flow disturbed by the initial blast wave. Then, the initial time t_i is determined using the location of the blast wave front. The coordinates of boundary meshes, x_b, y_b, and z_b should also be determined. Afterwards, both boundary conditions, p(x_b, y_b, z_b, t), v(x_b, y_b, z_b, t), and ρ(x_b, y_b, z_b, t), and initial conditions, p(x, y, z, t_i), v(x, y, z, t_i), and ρ(x, y, z, t_i), calculated through Equations (1), (2) and (4), were applied to run the numerical simulation.

As described, the study implemented the numerical simulation approach using the CESE solver within LS-DYNA software. Boundary conditions for air were defined using the keyword *CESE_BOUNDARY_PRESCRIBED to set p(x_b, y_b, z_b, t), v(x_b, y_b, z_b, t), and ρ(x_b, y_b, z_b, t), while initial conditions were established via *CESE_INITIAL_OPTION_ELEMENT by inputting p(x, y, z, t_i), v(x, y, z, t_i), and ρ(x, y, z, t_i). To ensure simulation stability, numerical viscosity was incorporated into the model. Furthermore, a previously established re-weighting technique [40] was activated by specifying the variable IDLMT = 0 within the *CESE_CONTROL_LIMITER keyword card, which introduces non-physical parameters α and β. Air surrounding the gabled frames was modeled as an ideal gas, configured using the *CESE_EOS_IDEAL_GAS keyword to assign its material properties.

Table 1. Material properties of air employed in the present study.

Specific Heat at Constant Volume J/(kg·K)	Specific Heat at Constant Pressure J/(kg·K)	Initial Density kg/m3	Initial Pressure kPa
717.50	1004.50	1.29	101.32

lists the specific air material properties used for simulations within this study [24].

2.3. CFD Scheme Formulation

The general vector form of the basic equations of a compressible fluid employed in CFD under one-dimensional (1D) flow is given as

$${\displaystyle \frac{\partial U}{\partial t}}+{\displaystyle \frac{\partial F}{\partial x}}=0,$$

(5)

where the vector U is constituted of conservative variables and F is a function of U. The derivative of F with respect to time is calculated as

$${\left(F_t\right)}_j^n={\left({\displaystyle \frac{\partial F}{\partial U}}\right)}_j^n{\left(U_t\right)}_j^n,$$

(6)

where U_t is also a function of U. The CE/SE method solves Equation (5) based on the following formula:

$${\left(U\right)}_j^n=\left[{\left(U\right)}_{j-1/2}^{n-1/2}+{\left(U\right)}_{j+1/2}^{n-1/2}+{\left(S\right)}_{j-1/2}^{n-1/2}-{\left(S\right)}_{j+1/2}^{n-1/2}\right]/2,$$

(7)

where

$${\left(S\right)}_j^n={\displaystyle \frac{\Delta x}{4}}{\left(U_x\right)}_j^n+{\displaystyle \frac{\Delta t}{\Delta x}}{\left(F\right)}_j^n+{\displaystyle \frac{{\left(\Delta t\right)}^2}{4\Delta x}}{\left(F_t\right)}_j^n,$$

(8)

and the subscript j represents a spatial coordinate x_j and superscript n represents a temporal coordinate t_n. It should be mentioned here that Equations (5)–(8) can be extended beyond the 1D case for higher-dimensional problems by including additional dimensions and terms [13].

A number of methods based on the CE/SE scheme have been developed to calculate the spatial derivative of U when applying Equation (7) [27,41]. The CESE solver of LS-DYNA applies a scheme similar to the a-α scheme [40] for this purpose. To this end, the CESE solver defines the variable U in Equation (7) as

$${\left(U^{\prime }\right)}_{j\pm 1/2}^n={\left(U\right)}_{j\pm 1/2}^{n-1/2}+{\displaystyle \frac{\Delta t}{2}}{\left(U_t\right)}_{j\pm 1/2}^{n-1/2}.$$

(9)

The spatial derivative is calculated as

$${\left(U_{x\pm }\right)}_j^n=\pm {\displaystyle \frac{{\left(U^{\prime }\right)}_{j\pm 1/2}^n-{\left(U\right)}_j^n}{\Delta x/2}},$$

(10)

and a parameter related to the stability condition is defined as

$$\sigma ={\displaystyle \frac{1}{2\left|\nu \right|}},$$

(11)

where ν is the CFL number. To this is added the parameter α via the following function:

$${\eta }_{\pm }\left(x_{+},x_{-},\alpha \right)={\displaystyle \frac{{\left|x_{\pm }\right|}^{\alpha }}{\min \left({\left|x_{+}\right|}^{\alpha },{\left|x_{-}\right|}^{\alpha }\right)}}-1$$

(12)

along with the following weighted averaging coefficients:

$$w_{\pm }\left(x_{+},x_{-},\alpha \right)={\displaystyle \frac{1+\sigma {\eta }_{\mp }}{2+\sigma \left({\eta }_{+}+{\eta }_{-}\right)}}.$$

(13)

The solver then calculates the weighted averaging coefficients as follows:

$$W_{\pm }=w_{\pm }\left(U_{x+},U_{x-},\alpha \right).$$

(14)

This yields the following new numerical analogue:

$${\left({\mathbf{U}}_x^{\left(\mathrm{w}\right)}\right)}_j^n=W_{-}{\left(U_{x-}\right)}_j^n+W_{+}{\left(U_{x+}\right)}_j^n.$$

(15)

Applying the parameter β to define a linear combination of this weighted average and the central-difference yields the following expression:

$${\left(U_x\right)}_j^n={\left({\mathbf{U}}_x^{\left(\mathrm{c}\right)}\right)}_j^n+\beta \left[{\left(U_x^{\left(\mathrm{w}\right)}\right)}_j^n-{\left(U_x^{\left(\mathrm{c}\right)}\right)}_j^n\right].$$

(16)

Finally, substituting the value of ${\left(U_x\right)}_j^n$ calculated by Equation (16) into Equation (7) solves the compressible fluid problem at mesh point (x_j, t_n).

It should be noted that the numerical solution of the governing equations for compressible ideal air—as obtained, for example, via the CE/SE scheme—provides the spatial and temporal distributions of air pressure, particle velocity, and density. The internal energy can subsequently be derived from pressure and density, enabling the temperature field to be determined. Therefore, the highly efficient numerical approach presented herein is fully capable of predicting temperature variations in blast simulations. However, given that the primary focus of this study is on the mechanical blast load (overpressure and impulse) for structural response analysis, and considering that the thermal effects typically have a negligible impact on the short-term dynamic response of structures subjected to far-field explosions, the explicit presentation and discussion of temperature results are not included herein.

As presented in Reference [14], the highly efficient numerical approach for predicting blast loads demonstrates strong agreement with both empirical data from the UFC guidelines and experimental measurements. The maximum relative error of the proposed method is less than 10%, making it as accurate as the LBE method and more accurate than the mapping method. In numerical simulations of blast wave propagation and their interaction with building structures in the far field—under scenarios involving a TNT equivalent mass of 250 t at various standoff distances—the proposed approach effectively predicts the distribution of peak overpressure on the target building surface. In terms of computational efficiency, the mapping method requires nearly five times the CPU time of the highly efficient numerical approach, while the LBE method consumes approximately 1.8 times the CPU time. Thus, the proposed method performs more efficiently than both the LBE and mapping methods, achieving accuracy comparable to LBE and superior to that of the mapping method. For further details regarding the numerical methodology, readers are referred to Reference [14].

3. Numerical Dissipation

3.1. Parameter Analysis

A free air burst explosion from a spherical explosive charge can be modeled as a 1D problem in spherical coordinates. An analysis of the impact of parameters α, β, and CFL on the level of numerical dissipation of the previously proposed numerical method solved in conjunction with the CE/SE scheme is conducted herein based on a free air burst explosion of 1 kg TNT, where the boundary of the spatial domain is 1.50 m away from explosive charge and the initial time is the moment when the shock wave arrives at a point where the distance from the blast origin is 1.80 m. The computational domain adopts a quarter-annulus geometry, discretized using a structured (mapped) mesh configuration. The peak incident overpressures obtained at a scaled distance of 2.03 m/kg^1/3 are compared under the different parameter values listed in

Table 2. Peak overpressures obtained under different parameter values.

Case	α	β	CFL	Overpressure (kPa)
1	0.1	1.0	0.50	215.47
2	2.0	1.0	0.50	190.69
3	5.0	1.0	0.50	187.47
4	10.0	0.0	0.05	191.85
5	10.0	0.5	0.05	185.35
6	10.0	1.0	0.05	181.84
7	2.0	1.0	0.10	188.60
8	2.0	1.0	0.05	187.09

.

Based on the data in Table 2, Figure 3 presents the influence of the non-physical parameters (α, β, CFL) on the calculated peak overpressure.

The results in Table 2 and Figure 3 demonstrate that the peak overpressure decreases when α and β increase individually with all other parameters held fixed. For example, the overpressure decreases by 13.0% when α increases from 0.1 to 5.0, and the overpressure decreases by 1.89% when β increases from 0.5 to 1.0, with all other parameters held fixed. This represents a positive correlation between these parameters and the level of numerical dissipation, which is also denoted as numerical viscosity. However, the overpressure increases by 1.92% when CFL increases from 0.05 to 0.50, with all other parameters held fixed. The increasing CFL number results in a decreasing numerical dissipation, presenting a negative correlation. The engineering significance of these variations lies in their direct impact on blast load assessment for structures. The peak overpressure is a primary parameter—along with impulse—that defines a blast load’s destructive potential. It plays a predominant role in determining the structural response, including deformation, damage level, and potential failure. Consequently, a discrepancy of 13% in the predicted peak overpressure—resulting solely from different choices of non-physical parameters—may lead to significantly different, and potentially non-conservative, assessments of structural safety. This underscores the importance of the guidelines provided in this study for selecting appropriate values of α, β, and CFL to ensure both computational efficiency and predictive accuracy in engineering applications.

3.2. Analysis Based on a Simplified Continuity Equation

The continuity equation is simplified into a 1D conventional equation when the velocity is constant. This equation is the simplest hyperbolic partial differential equation. The impact of parameters α, β, and CFL on the level of numerical dissipation of the previously proposed numerical method were subjected to further detailed analysis based on the following simplified 1D continuity equation for the density with a single unknown variable:

$${\displaystyle \frac{\partial u}{\partial t}}+a{\displaystyle \frac{\partial u}{\partial x}}=0,$$

(17)

where u is the density of air and a is denoted as the convection velocity, which represents the constant velocity of an air particle.

To isolate and analyze the intrinsic numerical dissipation of the CE/SE scheme, independent of the complexities of multi-dimensional flow and shock physics, a simplified one-dimensional linear convection equation (Equation (17)) is employed. This model problem, while simplistic, preserves the fundamental hyperbolic conservation law form of the governing Euler equations (Equation (5)). This approach is a standard methodology in numerical analysis for deriving the core properties of a computational scheme (e.g., [26]). The insights gained from this analysis are shown in the following sections to be consistent with results from full numerical simulations, confirming their value in guiding parameter selection for practical applications.

The specific scheme used to construct the analogue of ∂u/∂x involves defining the following terms.

$${\left(u^{\prime }\right)}_{j\pm 1/2}^n={\left(u\right)}_{j\pm 1/2}^{n-1/2}+{\displaystyle \frac{\Delta t}{2}}{\left(u_t\right)}_{j\pm 1/2}^{n-1/2}$$

(18)

$${\left(u_{x\pm }\right)}_j^n=\pm {\displaystyle \frac{{\left(u^{\prime }\right)}_{j\pm 1/2}^n-{\left(u\right)}_j^n}{\Delta x/2}}$$

(19)

$${\left(u_x^{\left(\mathrm{c}\right)}\right)}_j^n={\displaystyle \frac{{\left(u^{\prime }\right)}_{j+1/2}^n-{\left(u^{\prime }\right)}_{j-1/2}^n}{\Delta x}}$$

(20)

For convenience, the CFL number (ν) is now expressed as

$$\nu ={\displaystyle \frac{a\Delta t}{\Delta x}},$$

(21)

and the following two parameters are introduced:

$$\sigma ={\displaystyle \frac{1}{2\left|\nu \right|}},$$

(22)

$${\eta }_{\pm }={\displaystyle \frac{{\left|u_x^{\pm }\right|}^{\alpha }}{\min \left({\left|u_x^{+}\right|}^{\alpha },{\left|u_x^{-}\right|}^{\alpha }\right)}}-1,$$

(23)

where the corresponding weighted averaging coefficients are

$$w_{\pm }={\displaystyle \frac{1+\sigma {\eta }_{\mp }}{2+\sigma \left({\eta }_{+}+{\eta }_{-}\right)}}.$$

(24)

Then, a new numerical analogue is developed in the form of

$${\left(u_x^{\left(\mathrm{w}\right)}\right)}_j^n=w_{-}{\left(u_{x-}\right)}_j^n+w_{+}{\left(u_{x+}\right)}_j^n,$$

(25)

and the linear combination corresponding to Equation (16) is obtained as

$${\left(u_x\right)}_j^n={\left(u_x^{\left(\mathrm{c}\right)}\right)}_j^n+\beta \left[{\left(u_x^{\left(\mathrm{w}\right)}\right)}_j^n-{\left(u_x^{\left(\mathrm{c}\right)}\right)}_j^n\right].$$

(26)

The value of ${\left(u_x\right)}_j^n$ in Equation (26) is then substituted for the actual gradient value at mesh point (x_j, t_n).

Note that governing Equation (17) and its partial derivatives support the following relationships.

$$u_t=-a{u}_x$$

(27)

$$u_{tt}=a^2{u}_{xx}$$

(28)

$$u_{xt}=-a{u}_{xx}$$

(29)

Here, u_t represents ∂u/∂t, u_x represents ∂u/∂x, u_tt represents ∂²u/∂t², u_xx represents ∂²u/∂x², and u_xt represents ∂²u/(∂x∂t).

Substituting Equations (27)–(29) into Taylor series expansions of u and u_t around point (x_j, t_n) and ignoring 3rd order and higher-order terms yields the following:

$$\begin{array}{l}{\left(u\right)}_{j\pm 1/2}^{n-1/2}={\left(u\right)}_j^n\pm {\displaystyle \frac{\Delta x}{2}}{\left(u_x\right)}_j^n+{\displaystyle \frac{a\Delta t}{2}}{\left(u_x\right)}_j^n+\\ {\displaystyle \frac{\Delta x^2}{8}}{\left(u_{xx}\right)}_j^n\pm {\displaystyle \frac{a\Delta x\Delta t}{4}}{\left(u_{xx}\right)}_j^n+{\displaystyle \frac{a^2\Delta t^2}{8}}{\left(u_{xx}\right)}_j^n\dots \end{array}$$

(30)

and

$${\left(u_t\right)}_{j\pm 1/2}^{n-1/2}=-a\left({\left(u_x\right)}_j^n+\left({\displaystyle \frac{a\Delta t}{2}}\pm {\displaystyle \frac{\Delta x}{2}}\right){\left(u_{xx}\right)}_j^n\right)\dots$$

(31)

Substituting Equations (30) and (31) into Equations (18)–(20) and Equation (25) yields the following:

$${\left(u^{\prime }\right)}_{j\pm 1/2}^n={\left(u\right)}_j^n\pm {\displaystyle \frac{\Delta x}{2}}{\left(u_x\right)}_j^n+\left({\displaystyle \frac{\Delta x^2}{8}}-{\displaystyle \frac{a^2\Delta t^2}{8}}\right){\left(u_{xx}\right)}_j^n\dots$$

(32)

$${\left(u_{x\pm }\right)}_j^n={\left(u_x\right)}_j^n\pm \left({\displaystyle \frac{\Delta x}{4}}-{\displaystyle \frac{a^2\Delta t^2}{4\Delta x}}\right){\left(u_{xx}\right)}_j^n\dots$$

(33)

$${\left(u_x^{\left(\mathrm{c}\right)}\right)}_j^n={\left(u_x\right)}_j^n+\dots$$

(34)

$${\left(u_x^{\left(\mathrm{w}\right)}\right)}_j^n=\left(w_{+}+w_{-}\right){\left(u_x\right)}_j^n+\left(w_{+}-w_{-}\right)\left({\displaystyle \frac{\Delta x}{4}}-{\displaystyle \frac{a^2\Delta t^2}{4\Delta x}}\right){\left(u_{xx}\right)}_j^n\dots$$

(35)

Note that Equations (23) and (24) lead to

$$w_{+}+w_{-}=1.$$

(36)

Thus,

$${\left(u_t\right)}_j^n+a{\left(u_x\right)}_j^n=a\beta \left(w_{+}-w_{-}\right)\left({\displaystyle \frac{\Delta x}{4}}-{\displaystyle \frac{a^2\Delta t^2}{4\Delta x}}\right){\left(u_{xx}\right)}_j^n.$$

(37)

Considering the definition of the CFL number and ignoring scripts yields the following expression.

$${\displaystyle \frac{\partial u}{\partial t}}+a{\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{a}{4}}\beta \Delta x\left(w_{+}-w_{-}\right)\left(1-CF{L}^2\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}$$

(38)

The fact that the continuity equation given by Equation (17) is the origin of Equation (38) indicates that the scheme developed to calculate the spatial derivative of the unknown variable u in the CESE solver of LS-DYNA contains a numerical dissipation term whose coefficient is aβΔx(w₊ − w₋)(1 − CFL²)/4. Hence, the non-physical parameter α influences the degree of numerical dissipation by affecting the weighted averaging coefficients w₊/w₋.

The impacts of the parameters β, CFL, and mesh size on the particle velocities calculated by the simplified numerical model of the free air burst explosion introduced earlier in this section were analyzed at a moment when t > t_A(r), and the results are presented in Figure 4.

As can be seen, the particle velocity exhibits an inverse relationship with respect to β and mesh size, indicating that the degree of numerical dissipation decreases as these parameters decrease in magnitude. However, the numerical dissipation effect initially increases rapidly with an increasing CFL, but the increasing tendency diminishes with further increase in CFL beyond a value of about 0.5, showing a nonlinear relationship. These conclusions are consistent with the trends indicated by an analysis of the above-defined coefficient aβΔx(w₊ − w₋)(1 − CFL²)/4 of the numerical dissipation term in Equation (38).

4. Stability Analysis

The impacts of important CE/SE scheme parameters on the stability of the previously proposed numerical method were analyzed by means of von Neumann analysis [26]. Von Neumann analysis assesses stability by examining the growth of Fourier modes in the numerical solution. Here, we derive the amplification matrix for the scheme and impose the condition that its spectral radius (magnitude of largest eigenvalue) must be less than 1 for stability. For convenience, the analysis was again conducted based on the simplified continuity equation for the particle velocity in Equation (17).

To this end, we first make the following simplification more concise by defining the term:

$${\left({\overline{u}}_x\right)}_j^n={\displaystyle \frac{\Delta x}{4}}{\left(u_x\right)}_j^n.$$

(39)

The CFD formulation scheme used can be expressed according to the following two terms.

$$\begin{array}{l}{\left(u\right)}_j^n={\displaystyle \frac{1}{2}}\left(1+\nu \right){\left(u\right)}_{j-1/2}^{n-1/2}+{\displaystyle \frac{1}{2}}\left(1-{\nu }^2\right){\left({\overline{u}}_x\right)}_{j-1/2}^{n-1/2}+\\ {\displaystyle \frac{1}{2}}\left(1-\nu \right){\left(u\right)}_{j+1/2}^{n-1/2}-{\displaystyle \frac{1}{2}}\left(1-{\nu }^2\right){\left({\overline{u}}_x\right)}_{j+1/2}^{n-1/2}\end{array}$$

(40)

$$\begin{array}{l}{\left({\overline{u}}_x\right)}_j^n={\displaystyle \frac{1}{4}}\left(\beta \nu \left(1-2w_{+}\right)-1\right){\left(u\right)}_{j-1/2}^{n-1/2}+\\ \left({\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{\beta }{4}}\left(1-2w_{+}\right)\left(-{\nu }^2+2\nu +1\right)\right){\left(u_x\right)}_{j-1/2}^{n-1/2}+\\ {\displaystyle \frac{1}{4}}\left(-\beta \nu \left(1-2w_{+}\right)+1\right){\left(u\right)}_{j+1/2}^{n-1/2}+\\ \left(-{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{\beta }{4}}\left(1-2w_{+}\right)\left({\nu }^2+2\nu -1\right)\right){\left({\overline{u}}_x\right)}_{j+1/2}^{n-1/2}\end{array}$$

(41)

These equations are combined into a vector form for the solution state at point (j, n) which depends linearly on the solution states at points (j ± 1/2, n − 1/2). The solution vector is then

$$q\left(j,n\right)=\left(\begin{array}{c}{u}_j^n\\ {\left({\overline{u}}_x\right)}_j^n\end{array}\right).$$

(42)

Defining the following two matrices:

$$M^{+}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1+\nu & 1-{\nu }^2\\ -w_{-}+{\displaystyle \frac{1}{2}}\left(w_{-}-w_{+}\right)\left(1+\nu \right)& 2\nu w_{-}+{\displaystyle \frac{1}{2}}\left(w_{-}-w_{+}\right)\left(1-{\nu }^2\right)\end{array}\right)$$

(43)

and

$$M^{-}={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1-\nu & {\nu }^2-1\\ w_{+}+{\displaystyle \frac{1}{2}}\left(w_{-}-w_{+}\right)\left(1-\nu \right)& -2\nu w_{+}-{\displaystyle \frac{1}{2}}\left(w_{-}-w_{+}\right)\left(1-{\nu }^2\right)\end{array}\right)$$

(44)

enables the solution vector to be expressed as

$$q\left(j,n\right)=M^{+}q\left(j-1/2,n-1/2\right)+M^{-}q\left(j+1/2,n-1/2\right).$$

(45)

The solutions to Equation (45) are then approximated at a specific time as Fourier series. Substituting these approximations into Equation (45) yields the following forward marching operator for this scheme:

$$M=e^{-i\theta /2}{M}^{+}+e^{i\theta /2}{M}^{-},$$

(46)

where θ is the phase angle.

Equation (46) can be expressed in the form of the following trigonometric function:

$$M=\left(\begin{array}{cc}\cos \left(\theta /2\right)-\nu \sin \left(\theta /2\right)i& \sin \left(\theta /2\right)\left({\nu }^2-1\right)i\\ \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta \nu \left(1-2w_{+}\right)}{2}}\right)\sin \left(\theta /2\right)i& m_{22}\end{array}\right),$$

(47)

where

$$m_{22}=\beta \nu \left(1-2w_{+}\right)\cos \left(\theta /2\right)+\left({\displaystyle \frac{\beta \left({\nu }^2-1\right)\left(1-2w_{+}\right)}{2}}-\nu \right)\sin \left(\theta /2\right)i.$$

(48)

Chang [26] proved that necessary condition for the CFD formulation scheme to be stable follows from the spectral radius as follows:

$$r\left(M\right)\le 1.$$

(49)

Condition (49) must hold for all phase angles θ ∈ [0, 2π] to guarantee stability independent of the solution wavelength.

Computing the spectral radius requires finding the eigenvalues λ of the 2 × 2 matrix M. For the 2 × 2 matrix M, its spectral radius can be defined according to its two eigenvalues λ₁ and λ₂ as follows:

$$r\left(M\right)=\max \left\{\left|{\lambda }_1\right|,\left|{\lambda }_2\right|\right\}.$$

(50)

The eigenvalue calculation involves solving the characteristic equation of M, leading to complex expressions. Terms X and Y are introduced to organize the expressions arising from the eigenvalue calculation to further simplify the analysis. We define

$$\begin{array}{l}X=\left[{\displaystyle \frac{1}{8}}{\left({\nu }^2+1\right)}^2{\left(1-2w_{+}\right)}^2{\beta }^2-{\nu }^3\left(1-2w_{+}\right)\beta +{\nu }^2-{\displaystyle \frac{1}{2}}\right]\cos \theta \\ -{\displaystyle \frac{1}{8}}\left({\nu }^4+6{\nu }^2+1\right){\left(1-2w_{+}\right)}^2{\beta }^2+\nu \left({\nu }^2-2\right)\left(1-2w_{+}\right)\beta -{\nu }^2+{\displaystyle \frac{3}{2}}\end{array}$$

(51)

and

$$Y={\displaystyle \frac{1}{2}}\beta \left(1-{\nu }^2\right)\left(1-2w_{+}\right)\left[-\beta \left(1-2w_{+}\right)\nu +1\right]\sin \theta ,$$

(52)

and then calculate the eigenvalues of M as follows.

$$\begin{array}{l}{\lambda }_{1,2}={\displaystyle \frac{1}{2}}\left[\beta \nu \left(1-2w_{+}\right)+1\right]\cos \left(\theta /2\right)-\\ {\displaystyle \frac{1}{4}}\left[\beta \left(1-2w_{+}\right)\left(1-{\nu }^2\right)+4\nu \right]\sin \left(\theta /2\right)i\mp {\displaystyle \frac{1}{2}}\sqrt{X+Yi}\end{array}$$

(53)

Coefficients a and b (Equations (54) and (55)) represent the real and imaginary parts of a dominant term in the eigenvalue expression.

$$a={\displaystyle \frac{1}{2}}\left[\beta \nu \left(1-2w_{+}\right)+1\right]\cos \left(\theta /2\right)$$

(54)

$$b=-{\displaystyle \frac{1}{4}}\left[\beta \left(1-2w_{+}\right)\left(1-{\nu }^2\right)+4\nu \right]\sin \left(\theta /2\right)$$

(55)

Directly enforcing max|λ| ≤ 1 for all θ is complex. Instead, equivalent stability conditions are derived using auxiliary functions D, F and H (Equations (56)–(58)), which are constructed from X, Y, a and b. These functions are proven to be related to the magnitudes of the eigenvalues (Equations (59)–(61)).

The formalism is then applied to simplify the necessary condition for stability expressed in Equation (49) based on the following three functions.

$$D={\displaystyle \frac{1}{8}}\left(X^2+Y^2\right)-X\left(a^2-b^2\right)-2abY+2{\left(a^2+b^2-1\right)}^2$$

(56)

$$F=-\left(X^2+Y^2\right)+8X\left(a^2-b^2\right)+16\left[abY-{\left(a^2+b^2\right)}^2+1\right]$$

(57)

$$H={\left[{\displaystyle \frac{1}{8}}\left(X^2+Y^2\right)-X\left(a^2-b^2\right)-2abY+2{\left(a^2+b^2-1\right)}^2\right]}^2-\left(X^2+Y^2\right)$$

(58)

It can be proved that these three functions adhere to the following relationships.

$$D=2\left(1-{\left|{\lambda }_1\right|}^2\right)\left(1-{\left|{\lambda }_2\right|}^2\right)+\sqrt{X^2+Y^2}$$

(59)

$$F=16\left(1-{\left|{\lambda }_1\right|}^2{\left|{\lambda }_2\right|}^2\right)$$

(60)

$$H=D^2-\left(X^2+Y^2\right)$$

(61)

Therefore, the spectral radius condition in Equation (49) is mathematically equivalent to requiring the following equations hold simultaneously for all θ.

$$\left\{\begin{array}{c}D\ge 0\\ F\ge 0\\ H\ge 0\end{array}\right.$$

(62)

Expanding Equations (56)—and combining coefficients in terms of the power of cosθ yields the following expressions for D, F, and H.

$$D=a_D\left(\beta ,\nu ,w_{+}\right){\cos }^2\theta +b_D\left(\beta ,\nu ,w_{+}\right)\cos \theta +c_D\left(\beta ,\nu ,w_{+}\right)$$

(63)

$$F=a_F\left(\beta ,\nu ,w_{+}\right){\cos }^2\theta +b_F\left(\beta ,\nu ,w_{+}\right)\cos \theta +c_F\left(\beta ,\nu ,w_{+}\right)$$

(64)

$$\begin{array}{l}H=a_H\left(\beta ,\nu ,w_{+}\right){\cos }^4\theta +b_H\left(\beta ,\nu ,w_{+}\right){\cos }^3\theta +\\ c_H\left(\beta ,\nu ,w_{+}\right){\cos }^2\theta +d_H\left(\beta ,\nu ,w_{+}\right)\cos \theta +e_H\left(\beta ,\nu ,w_{+}\right)\end{array}$$

(65)

As can be seen, the coefficients involving the variables a, b, c, d, and e are functions of the parameters β, ν, and w₊, and the subscripts D, F, and H denote coefficients pertaining to the individual functions. Finally, Equations (63) and (64) can be simplified into quadratic functions of the form y = ax² + bx + c and Equation (65) can be simplified into a quartic function of the form y = ax⁴ + bx³ + cx² + dx + e with x∈[−1.0, 1.0] substituting for cosθ.

The results obtained from calculations involving a massive number of cases with different values of β, ν, and w₊ demonstrate that Equation (62) is true when the following inequalities are met.

$$\left\{\begin{array}{c}0\le \beta \le 1\\ 0\le w_{+}\le 1\\ 0\le \nu \le 1\end{array}\right.$$

(66)

Recalling that w_x depends on the parameter α (introduced in the spatial derivative calculation, Equation (12)), and that α ≥ 0 ensures 0 ≤ w_x ≤ 1, the stability conditions are equivalently expressed in terms of the primary scheme parameters α, β, CFL(ν). The inequalities can be expressed with as follows

$$\left\{\begin{array}{c}0\le \beta \le 1\\ \alpha \ge 0\\ 0\le CFL\le 1\end{array}\right.$$

(67)

Thus, Equation (67) is confirmed as the stability condition for the CE/SE scheme considered herein for calculating solutions of Equation (17). This provides clear, practical bounds for selecting the non-physical parameters α, β and CFL to ensure stable simulations. However, calculating solutions for Euler equations of inviscid compressible fluid to predict blast loads requires that the CFL number be defined as follows [27]:

$$CFL=\left(\left|\mathit{v}\right|+\left|c\right|\right){\displaystyle \frac{\Delta t}{\Delta x}}$$

(68)

where v is the vector air particle flow velocity and c is the speed of sound.

In practical numerical simulations of far-field blast loads, nonphysical parameters must strictly satisfy the stability conditions given in Equation (67). As shown by the numerical dissipation coefficient derived in Equation (38), larger values of β introduce greater numerical dissipation into the scheme, leading to increased inaccuracy. It is therefore recommended to minimize β. However, for flows involving large gradients or discontinuities such as shocks, β = 1.0 is highly recommended [40]. Consequently, β should be set to 1 for most blast load prediction scenarios. According to reference [40], α = 1 is typically sufficient for most cases, except when the shock is too strong, where a larger value may be required. For the CFL number, it is recommended to maintain a value below 0.9 to ensure stability in practical numerical simulations of blast load prediction.

5. Discussion

This study has investigated the numerical dissipation and stability of a highly efficient numerical approach, previously proposed for predicting blast loads from large TNT-equivalent explosives on building structures. The method incorporates a rapid algorithm to simulate airflow disturbed by blast shock waves. These flow behaviors are used to define initial and boundary conditions, thereby establishing an initial blast wave positioned in close proximity to the target structure. This strategy significantly reduces computational cost compared to the mapping method, as it eliminates the need to simulate one-dimensional detonation and subsequent shock propagation. Similarly, it offers improved efficiency over the LBE method by avoiding full simulation of shock wave travel from distant inlet boundaries.

The numerical approach is implemented using a CE/SE solver within the commercial software LS-DYNA. It is important to note that the specific numerical scheme employed here differs from other established CE/SE variants—such as the c-τ or c schemes, whose numerical properties have been extensively analyzed in earlier studies [26,27]. Therefore, the present paper focuses specifically on the dissipation and stability behavior of this LS-DYNA-integrated formulation.

It should be noted that the analysis of numerical dissipation presented in this paper is based on numerical simulations of free air burst scenarios and theoretical one-dimensional continuity equations. The conclusions, including the explicit formula for calculating the dissipation term, reveal the general trend regarding the numerical dissipation of the novel approach proposed for predicting blast loads. However, these findings and the associated formula are not directly applicable to quantifying numerical dissipation in other numerical simulation cases that employ this novel numerical approach.

The stability condition prescribed by Equation (67) is necessary for most scenarios. However, in certain specific simulation cases, a more restrictive stability condition may be required. The stability should be implicitly confirmed by examining the simulation results upon completion to ensure numerical reliability and accuracy.

Based on the findings presented in this study, several promising directions for future research emerge, building upon the numerical dissipation and stability characteristics identified for the LS-DYNA-integrated CE/SE formulation. From a scientific perspective, the relationship between non-physical parameters and numerical dissipation could be further generalized and quantitatively modeled across a wider variety of blast scenarios. Such efforts may facilitate the development of systematic compensation strategies, thereby enhancing the predictive accuracy of blast simulations without compromising computational efficiency. From an applied standpoint, the development of more comprehensive and universally applicable stability criteria represents a valuable direction. Refined stability conditions would reduce the need for iterative trial simulations and provide greater reliability across diverse engineering contexts, further increasing the practicality of the proposed method for real-world blast assessment and structural design.

6. Conclusions

This study was motivated by a practical need in computational blast dynamics: to provide clear guidance on the use of a highly efficient CE/SE-based method implemented in LS-DYNA, specifically by quantifying the impact of its non-physical parameters (α, β, CFL) on numerical dissipation and stability. Our primary aim was to establish an analytical framework to inform parameter selection, thereby enhancing the reliability and predictive accuracy of this approach.

The main outcomes of this work are summarized as follows:

(1)

Theoretical Framework: Based on a simplified 1D model, an explicit expression for the numerical dissipation term was derived (aβΔx(w₊ − w_-)(1 − CFL²)/4). This result provides a quantitative basis for understanding how the parameters govern numerical dissipation within the scheme.

(2)

Stability Criteria: Through von Neumann analysis, the necessary stability conditions for the scheme were rigorously established as α ≥ 0, 0 ≤ β ≤ 1, and 0 ≤ CFL ≤ 1, providing essential bounds for robust simulation setup.

(3)

Parametric Analysis: Numerical simulations confirmed the theoretical trends, demonstrating that numerical dissipation increases with parameters α and β but decreases with the CFL number.

In comparison to other conventional approaches for blast simulation, this method offers a compelling balance of efficiency and accuracy. As demonstrated in [14], it achieves a level of accuracy comparable to the LBE method while requiring only about 55% of the computational time. More strikingly, it requires only about 20% of the CPU time consumed by the resource-intensive Mapping method. This high computational efficiency makes it particularly advantageous for large-scale parametric studies or engineering design scenarios involving far-field blast loads.

The insights from this study furnish practical guidelines for users: employing β = 1 and α = 1 is recommended for most blast scenarios involving strong shocks to control numerical oscillations, while maintaining a CFL number below 0.9 is advised to ensure stability. In a broader context, the CE/SE scheme differs from conventional finite-volume or finite-element methods by solving integral conservation laws directly on unified space–time elements, which intrinsically suppresses numerical oscillations at shock fronts without relying on artificial viscosity. Therefore, the proposed approach and the guidelines presented herein are most suitable for predicting the precise peak overpressure and impulse from far-field, large-scale explosions on structures, where computational efficiency and capturing sharp shock fronts are paramount. Future work will aim to extend this analysis to more complex multi-dimensional scenarios.