Search Results (29)

Godunov-based finite volume (FV) methods are widely employed to numerically solve the Shallow-Water Equations (SWEs) with application to simulate flood inundation over irregular geometries and real-field, where unstructured triangular meshing is favored. Second-order extensions have been devised, mostly on the MUSCL reconstruction and the discontinuous Galerkin (DG) approaches. In this paper, we introduce a novel second-order Runge–Kutta discontinuous Galerkin (RKDG) solver for flood modeling, specifically addressing positivity preservation and wetting and drying on unstructured triangular meshes. To enhance the RKDG model, we adapt and refine positivity-preserving and wetting and drying techniques originally developed for the MUSCL-based finite volume (FV) scheme, ensuring its effective integration within the RKDG framework. Two analytical test problems are considered first to validate the proposed model and assess its performance in comparison with the MUSCL formulation. The performance of the model is further explored in real flooding scenarios involving irregular topographies. Our findings indicate that the added complexity of the RKDG model is justified, as it delivers higher-quality results even on very coarse meshes. This reveals that there is a promise in deploying RKDG-based flood models in real-scale applications, in particular when field data are sparse or of limited resolution. Full article

In numerical simulations of underwater explosions, inaccuracies in the parameters of the Jones–Wilkins–Lee (JWL) equation of state often result in significant deviations between predicted shock wave pressure peaks or bubble pulsation periods and experimental or empirical results. To achieve the precise forecasting of underwater explosion loads, a corrected method for adjusting the initial conditions of explosives is proposed. This method regulates explosion loads by correcting the initial density and initial internal energy per unit mass of the explosive, offering a straightforward implementation and easy extension to complex scenarios. In addition, the accuracy and feasibility of the proposed method were validated through comparisons with experimental data and empirical formulas from international studies. The numerical framework employs the Runge–Kutta Discontinuous Galerkin (RKDG) method to solve the one-dimensional Euler equations. The spatial discretization of the Euler domain is achieved using the discontinuous Galerkin (DG) method, while temporal discretization utilizes a third-order Runge–Kutta (RK) method. The results demonstrate that the proposed correction method effectively compensates for load discrepancies caused by inaccuracies in the JWL equation of state parameters. After correction, the maximum error in the shock wave pressure peak is reduced to less than 4.5%, and the maximum error in the bubble pulsation period remains below 1.9%. Full article

Solving the Euler equations often requires expensive computations of complex, high-order time derivatives. Although Taylor Discontinuous Galerkin (TDG) schemes are renowned for their accuracy and stability, directly evaluating third-order tensor derivatives can significantly reduce computational efficiency, particularly for large-scale, intricate flow problems. To overcome this difficulty, this paper presents an optimized numerical procedure that combines Taylor series time integration with the Discontinuous Galerkin (DG) approach. By replacing cumbersome tensor derivatives with simpler time derivatives of the Jacobian matrix and finite difference method inside the element to calculate the high-order time derivative terms, the proposed method substantially decreases the computational cost while maintaining accuracy and stability. After verifying its fundamental feasibility in one-dimensional tests, the optimized TDG method is applied to a two-dimensional forward-facing step problem. In all numerical tests, the optimized TDG method clearly exhibits a computational efficiency advantage over the conventional TDG method, therefore saving a great amount of time, nearly 70%. This concept can be naturally extended to higher-dimensional scenarios, offering a promising and efficient tool for large-scale computational fluid dynamics simulations. Full article

As an effective geophysical tool, ground penetrating radar (GPR) is widely used for environmental and engineering detections. Numerous numerical simulation algorithms have been developed to improve the computational efficiency of GPR simulations, enabling the modeling of complex structures. The discontinuous Galerkin method is a high efficiency numerical simulation algorithm which can deal with complex geometry. This method uses numerical fluxes to ensure the continuity between elements, allowing Maxwell’s equations to be solved within each element without the need to assemble a global matrix or solve large systems of linear equations. As a result, memory consumption can be significantly reduced, and parallel solvers can be applied at the element level, facilitating the construction of high-order schemes to enhance computational accuracy. In this paper, we apply the discontinuous Galerkin (DG) method based on unstructured meshes to 3D GPR simulation. To verify the accuracy of our algorithm, we simulate a full-space vacuum and a cuboid in a homogeneous medium and compare results, respectively, with the analytical solutions and those from the finite-difference method. The results demonstrate that, for the same error level, the proposed DG method has significant advantages over the FDTD method, with less than 20% of the memory consumption and calculation time. Additionally, we evaluate the effectiveness of our method by simulating targets in an undulating subsurface, and further demonstrate its capability for simulating complex models. Full article

This article presents a comparison between the performance obtained by using a spatial discretization of the Euler equations based on a high-order discontinuous Galerkin (dG) method and different sets of variables. The sets of variables investigated are as follows: (1) conservative variables; (2) primitive variables based on pressure and temperature; (3) primitive variables based on the logarithms of pressure and temperature. The solution is advanced in time by using a linearly implicit high-order Rosenbrock-type scheme. The results obtained using the different sets are assessed across several canonical unsteady test cases, focusing on the accuracy, conservation properties and robustness of each discretization. In order to cover a wide range of physical flow conditions, the test-cases considered here are (1) the isentropic vortex convection, (2) the Kelvin–Helmholtz instability and (3) the Richtmyer–Meshkov instability. Full article

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions. Full article

This paper investigates the dynamics of Richtmyer–Meshkov instability (RMI) in shocked backward-triangular bubbles through numerical simulations. Two distinct gases, He and ${\mathrm{SF}}_6$, are used within the backward-triangular bubble, surrounded by ${\mathrm{N}}_2$ gas. Simulations are conducted at two distinct strengths of incident shock wave, including ${\mathrm{M}}_s=1.25$ and 1.50. A third-order modal discontinuous Galerkin (DG) scheme is applied to simulate a physical conservation laws of two-component gas flows in compressible inviscid framework. Hierarchical Legendre modal polynomials are employed for spatial discretization in the DG platform. This scheme reduces the conservation laws into a semi-discrete set of ODEs in time, which is then solved using an explicit 3rd-order SSP Runge–Kutta scheme. The results reveal significant effects of bubble density and Mach numbers on the growth of RMI in the shocked backward-triangular bubble, a phenomenon not previously reported. These effects greatly influence flow patterns, leading to intricate wave formations, shock focusing, jet generation, and interface distortion. Additionally, a detailed analysis elucidates the mechanisms driving vorticity formation during the interaction process. The study also thoroughly examines these effects on the flow fields based on various integral quantities and interface characteristics. Full article

The unsteady combustion of solid propellants under oscillating environments is the key to understanding the combustion instability inside solid rocket motors. The discontinuous Galerkin–finite element method (DG-FEM) is introduced to provide an efficient yet flexible numerical platform to investigate the combustion dynamics of solid propellants. The algorithm is developed for the classical unsteady model, the Zel’dovich–Novozhilov model. It is then validated based on a special analytical solution. The DG-FEM algorithm is then compared with the classical spectral method based on Laguerre polynomials. It is shown that the DG-FEM works more efficiently than the traditional spectral method, providing a more accurate solution with a lower computational cost. Full article

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3). Full article

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of $O\left(h^{p+2}\right)$. Full article

Nonlinear coupled reaction–diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray–Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge–Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications. Full article

Overcoming the problem of excessive engine noise at low altitudes is a formidable task on the way to developing a supersonic passenger aircraft. The focus of this paper is on the fan noise shielding during take-off, investigated as part of the DLR project ELTON SST (estimation of landing and take-off noise of supersonic transport) for an in-house aircraft design. The supersonic inlet is required to provide the proper quantity and uniformity of air to the engine over a wider range of flight conditions than the subsonic inlet. For passenger aircraft, the noise problem influences engine integration and placement, and the new generation of supersonic transport would require innovative engineering solutions in order to come up with an efficient low-noise design. Potential solutions are evaluated using DLR tools capable of accurate source generation and noise propagation to the far-field. For low-speed aircraft operation, the method of choice is a strongly coupled volume-resolving discontinuous Galerkin (DG) and fast multipole boundary element method (FM-BEM) which is applied due to a large disparity between the Mach numbers on the interior and exterior of the inlet. The method is used for obtaining the acoustic signature of the full-scale model at realistic flight points, including the application of the programmed lapse rate (PLR), which involves simulations at higher pitch angles than for the reference flight path. The results show that the proposed method is highly suitable for obtaining accurate noise footprints during the low-speed phase and could be used to assist with certification procedures of future supersonic aircraft. Full article

Introduction: Lung Ultrasonography (LUS) is a fast technique for the diagnosis of patients with respiratory syndromes. B-lines are seen in response to signal reverberations and amplifications into sites with peripheral lung fluid concentration or septal thickening. Mathematical models are commonly applied in biomedicine to predict biological responses to specific signal parameters. Objective: This study proposes a Finite-Element numerical model to simulate radio frequency ultrasonic lines propagated from normal and infiltrated lung structures. For tissue medium, a randomized inhomogeneous data method was used. The simulation implemented in COMSOL^® used Acoustic Pressure and Time-Explicit models, which are based on the discontinuous Galerkin method (dG). Results: The RF signals, processed in MATLAB^®, resulted in images of horizontal A-lines and vertical B-lines, which were reasonably similar to real images. Discussion: The use of inhomogeneous materials in the model was good enough to simulate the scattering response, similar to others in the literature. The model is useful to study the impact of the lung infiltration characteristics on the appearance of LUS images. Full article

Multi-component flow problems are typical of many technological and engineering applications. In this work, we propose an implicit high-order discontinuous Galerkin discretization of the variable density incompressible (VDI) flow model for the simulation of multi-component problems. Indeed, the peculiarity of the VDI model is that the density is treated as an advected property, which can be used to possibly track multiple (more than two) components. The interface between fluids is described by a smooth, but sharp, variation in the density field, thus not requiring any geometrical reconstruction. Godunov numerical fluxes, density positivity, mass conservation, and Gibbs-type phenomena at material interfaces are challenges that are considered during the numerical approach development. To avoid Courant-related time step restrictions, high-order single-step multi-stage implicit schemes are applied for the temporal integration. Several test cases with known analytical solutions are used to assess the current approach in terms of space, time, and mass conservation accuracy. As a challenging application, the simulation of a 2D droplet impinging on a thin liquid film is performed and shows the capabilities of the proposed DG approach when dealing with high-density (water–air) multi-component problems. Full article

A high-order discontinuous Galerkin (DG) method is presented for solving the preconditioned Euler equations with an explicit or implicit time marching scheme. A detailed description is given of a practical implementation of a precondition matrix of the type of Weiss and Smith and of the DG spatial discretization scheme employed, with particular emphasis on the artificial viscosity-based shock capturing techniques. The curved boundary treatment is proposed through adopting a NURBS surface equipped with a radial basis function interpolation to propagate the boundary displacement to the interior of the mesh. The resulting methods are verified by simulating flows over two-dimensional airfoils, such as symmetric NACA0012 or asymmetric RAE2822, and over three-dimensional bodies, such as an academic hemispherical headform or aerodynamic ONERA M6 wing. Numerical results show that the present method functions for both transonic and nearly incompressible flow simulations, and the proposed treatment of curved boundaries, play an important role in improving the accuracy of the obtained solutions, which are in good agreement with available experimental data or other numerical solutions reported in literature. Full article