A Multiple-Scale Space–Time Collocation Trefftz Method for Two-Dimensional Wave Equations

Abstract

This paper presents a semi-analytical, mesh-free space–time Collocation Trefftz Method (SCTM) for solving two-dimensional (2D) wave equations. Given prescribed initial and boundary data, collocation points are placed on the space–time (ST) boundary, reformulating the initial value problem as an equivalent boundary value problem and enabling accurate reconstruction of wave propagation in complex domains. The main contributions of this work are twofold: (i) a unified ST Trefftz basis that treats time as an analytic variable and enforces the wave equation in the full ST domain, thereby eliminating time marching and its associated truncation-error accumulation; and (ii) a Multiple-Scale Characteristic-Length (MSCL) grading strategy that systematically regularizes the collocation linear system. Several numerical examples, including benchmark tests, validate the method’s feasibility, effectiveness, and accuracy. For both forward and inverse problems, the solutions produced by the method closely match exact results, confirming its accuracy. Overall, the results reveal the method’s feasibility, accuracy, and stability across both forward and inverse problems and for varied geometries.

1. Introduction

The wave equation is a fundamental mathematical model that represents diverse wave phenomena, such as sound, electromagnetic, seismic, and water waves [1]. Its study facilitates the understanding of the propagation mechanisms and behavioral characteristics of these phenomena, enabling effective prediction of their impacts.

In practical engineering problems, deriving analytical solutions are challenging due to the complex boundary conditions and the geometric nature of computational domains [2]. Therefore, numerical simulations of accurate solutions are crucial. Currently, numerical solutions primarily employ classical mesh methods, such as the Finite Difference Method (FDM) [3], finite volume method [4], boundary finite element method [5], and Finite Element Method (FEM) [6]. However, these mesh-based approaches struggle to construct appropriate meshes to facilitate the analysis in complex computational domains, resulting in discretization errors; therefore, they are generally limited to simpler domains. To overcome these challenges, hybrid formulations have been developed. They couple the boundary element method with FEM [7] to handle wave propagation in complex layered structures, or with the curved virtual element method [8] to simulate wave fields in unbounded or geometrically complex domains. In addition, modern approaches, such as isogeometric analysis [9], have been introduced. They significantly alleviate the difficulties of mesh generation and extend the applicability of mesh-based methods to complex geometries. Nevertheless, meshless methods provide additional advantages, including eliminating mesh generation, allowing flexible placement of collocation points, and offering high computational efficiency; this makes them particularly suitable for irregular domains [10]. For instance, some solutions are directly implemented, such as the employment of the localized radial basis function collocation method to analyze wave propagation [11], or the achievement of high-precision solutions through semi-analytical spatial discretization methods, such as the Trefftz method [12], Method of Particular Solutions (MPS) [13], and Method of the Fundamental Solution (MFS) [14]. Notably, most current numerical methods still rely on the FDM for handling time variables, while other numerical techniques are utilized for the remaining processes [15]. However, wave problems require the calculation of second-order time derivatives, and the use of these numerical methods can lead to cumulative errors over time due to truncation errors, thus limiting the precision of solutions [16]. Consequently, the time discretization of second-order time derivatives and the spatial discretization of irregular domains are significant influencing factors that affect the accuracy of these types of numerical solutions for wave equations.

Beyond conventional numerical strategies, deep learning-based approaches have also been investigated for analyzing wave simulations [17,18], where physical modeling is embedded within the learning process. In contrast, our method directly employs Trefftz basis functions that exactly satisfy the governing equations, thereby achieving greater accuracy and robustness than purely data-driven networks.

This study formulates a Trefftz framework within a genuinely one-step ST Collocation setting, enabling the direct solution of wave problems without time marching. To ensure accuracy in irregular spatial domains, the proposed Space–time Collocation Trefftz Method (SCTM) [19] adopts T-complete functions as basis solutions of the wave equation. Although these functions theoretically guarantee high accuracy and exponential convergence, the linear algebraic system derived from SCTM is highly ill-conditioned, leading to suboptimal numerical results. To address this issue, the study introduces Multiple-Scale Characteristic-Length (MSCL) strategy during the solution process, enhancing both system stability and computational accuracy.

The core innovation of this work lies in combining the strengths of Trefftz basis functions, MSCL scaling, and a direct ST collocation strategy into a unified framework. Specifically, the proposed ST collocation multiple-scale Trefftz method simplifies the traditional two-step solution process by employing a consistent discretization in both space and time, using an ST mesh-free collocation approach for solving partial differential equations. Unlike conventional two-step FDM-based methods, this one-step direct mesh-free approach eliminates truncation-error accumulation, improves robustness against ill-conditioning, and achieves higher computational efficiency.

The rest of this paper is structured as follows: Section 2 elaborates on the mathematical foundations of the wave equation and outlines the methodology for addressing the wave equation under initial and boundary conditions. Section 3 illustrates the stability and accuracy of the proposed method. Several benchmark examples are provided to assess the performance of the developed method, as detailed in Section 4. In Section 5, the conclusions and key findings of this study are discussed. Moreover, some directions for future work will be provided.

2. Numerical Method

2.1. The Wave Equation

The wave equation is fundamental in many areas of physics and engineering. The wave equation is generally expressed as follows:

$${\displaystyle \frac{{\mathsf{\partial }}^{2}u}{{\mathsf{\partial }}^{2}t}}=a^2{\nabla }^{2}u,$$

(1)

where ${\nabla }^2$ is the Laplace operator, $u$ denotes the wave function, which can represent displacement, pressure, electric field intensity, among others, $t$ represents the time, and $a$ highlights the wave speed. In the ST domain, the two-dimensional (2D) wave equation is expressed as follows:

$${\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2}}=a^2\left({\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{r}^2}}+{\displaystyle \frac{\mathsf{\partial }u}{r\mathsf{\partial }r}}+{\displaystyle \frac{{\mathsf{\partial }}^{2}u}{r^2\mathsf{\partial }{\theta }^2}}\right), \left(r,\theta ,t\right)\in {\mathsf{\Omega }}_t,$$

(2)

where $r$ represents the radius, $\theta$ denotes the polar angle, and ${\mathsf{\Omega }}_t$ is the ST domain. The initial conditions are defined as follows:

$$u=u_0(r, \theta , 0), \left(r,\theta ,0\right)\in {\mathsf{\Omega }}_t,$$

(3)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}={\displaystyle \frac{\mathsf{\partial }{u}_0(r, \theta , 0)}{\mathsf{\partial }t}},\left(r,\theta ,0\right)\in {\mathsf{\Omega }}_t,$$

(4)

where $u_0$ is the initial value. The boundary condition is defined as follows:

$$u=F_D(r, \theta , t)\mathrm{on} {\mathsf{\Gamma }}^D,$$

(5)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }n}}=F_N(r, \theta , t)\mathrm{on} {\mathsf{\Gamma }}^N,$$

(6)

where $n$ indicates the outward normal direction, $F_D(r, \theta , t)$ supplies the Dirichlet data, and $F_N(r, \theta , t)$ provides the Neumann data. In terms of boundary conditions, ${\mathsf{\Gamma }}^D$ and ${\mathsf{\Gamma }}^N$ refer to Dirichlet and Neumann boundary segments, respectively.

2.2. Space–Time Collocation Trefftz Method

The SCTM is a mesh-free numerical technique for solving transient boundary-value problems. This collocation scheme is employed to solve wave propagation issues in irregular 2D spatial regions, as shown in Figure 1a. Through the application of the ST collocation scheme, the transformation extends the spatial dimensions into a combined space–time framework, as illustrated in Figure 1b. This approach allows for the simultaneous consideration of both spatial and temporal variables, thereby enhancing the solution accuracy by capturing the dynamic interactions between space and time. Within this ST domain, boundary conditions and extended time axes at collocation points are available, treating initial conditions equivalently as boundary conditions. Figure 1 depicted the initial and boundary collocation points within the ST domain for a 2D transient problem. The indirect Trefftz formulation involves selecting a set of basic functions that inherently fulfill the governing differential equation. This approach involves separating the variable method to obtain a set of T-type functions, listed in

Table 1. The basis for the T-complete functions.

Index	Formula	Index	Formula
${\mathrm{T}}_1$	$1$	${\mathrm{T}}_{10}$	$\sin (Hat)J_0\left(Hr\right)$
${\mathrm{T}}_2$	$t$	${\mathrm{T}}_{11}$	$I_G\left(Hr\right)\cos (G\theta )e^{Hat}$
${\mathrm{T}}_3$	$r^G\cos \left(G\theta \right)$	${\mathrm{T}}_{12}$	$I_G\left(Hr\right)\sin (G\theta )e^{Hat}$
${\mathrm{T}}_4$	$r^G\sin (G\theta )$	${\mathrm{T}}_{13}$	$I_G\left(Hr\right)\cos (G\theta )e^{-Hat}$
${\mathrm{T}}_5$	$t{r}^G\sin (G\theta )$	${\mathrm{T}}_{14}$	$I_G\left(Hr\right)\sin (G\theta )e^{-Hat}$
${\mathrm{T}}_6$	$t{r}^G\cos \left(G\theta \right)$	${\mathrm{T}}_{15}$	$\cos (Hat)\cos (G\theta )J_G(Hr)$
${\mathrm{T}}_7$	$e^{Hat}{I}_0(Hr)$	${\mathrm{T}}_{16}$	$\cos (Hat)\sin (G\theta )J_G(Hr)$
${\mathrm{T}}_8$	$e^{-Hat}{I}_0(Hr)$	${\mathrm{T}}_{17}$	$\sin (Hat)\cos (G\theta )J_G(Hr)$
${\mathrm{T}}_9$	$\cos (Hat)J_0\left(Hr\right)$	${\mathrm{T}}_{18}$	$\sin (Hat)\sin (G\theta )J_G(Hr)$

$J_0$: Bessel function of the first kind. $J_G$: Bessel function of the first kind of order $G$. $I_0$: Modified Bessel functions of the first kind. $I_G$: Modified Bessel functions of the first kind of order $G$. $H$ and $G$: The order of the basis functions, which are arbitrary real numbers.

. The computed approximation at each collocation point is subsequently expressed as a linear combination of these basis functions, with additional unknown functions or coefficients introduced to ensure that the solution meets the boundary conditions and accurately represents the underlying physical phenomenon. The SCTM starts by considering T-complete functions, being crucial as they enable the formulation of solutions across simply connected, infinite, doubly connected, and multiply connected domains. For a simply connected domain, the solution can be obtained using only the positive basis functions, expressed as follows:

$$\begin{array}{l}u(r,\theta ,t)=A^1{T}_1+A^2{T}_2+{\displaystyle \sum _{g=1}^m\left(A_g^3{T}_3+A_g^4{T}_4+A_g^5{T}_5+A_g^6{T}_6+A_g^7{T}_7+A_g^8{T}_8\right)}+\\ {\displaystyle \sum _{h=1}^s\left(A_h^7{T}_7+A_h^8{T}_8+A_h^9{T}_9+A_h^{10}{T}_{10}\right)+}\\ {\displaystyle \sum _{h=1}^s{\displaystyle \sum _{g=1}^m\left(A_{hg}^{11}{T}_{11}+A_{hg}^{12}{T}_{12}+A_{hg}^{13}{T}_{13}+A_{hg}^{14}{T}_{14}+A_{hg}^{15}{T}_{15}+A_{hg}^{16}{T}_{16}+A_{hg}^{17}{T}_{17}+A_{hg}^{18}{T}_{18}\right)}}\end{array}\mathrm{in} {\mathsf{\Omega }}_t,$$

(7)

where $A^1,A^2,A_{\mathrm{g}}^3,\cdots ,A_{hg}^{18}$ represent arbitrary constants that need to be determined. The following results are obtained by implementing Neumann boundary condition:

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }u(r,\theta ,t)}{\mathsf{\partial }n}}=A^1{T}_1^{\prime }+A^2{T}_2^{\prime }+{\displaystyle \sum _{g=1}^m\left(A_g^3{T}_3^{\prime }+A_g^4{T}_4^{\prime }+A_g^5{T}_5^{\prime }+A_g^6{T}_6^{\prime }\right)}+\\ {\displaystyle \sum _{h=1}^s\left(A_h^7{T}_7^{\prime }+A_h^8{T}_8^{\prime }+A_h^9{T}_9^{\prime }+A_h^{10}{T}_{10}^{\prime }\right)}+\\ {\displaystyle \sum _{h=1}^s{\displaystyle \sum _{g=1}^m\left(\begin{array}{l}{A}_{hg}^{11}{T}_{11}^{\prime }+A_{hg}^{12}{T}_{12}^{\prime }+A_{hg}^{13}{T}_{13}^{\prime }+A_{hg}^{14}{T}_{14}^{\prime }+\\ A_{hg}^{15}{T}_{15}^{\prime }+A_{hg}^{16}{T}_{16}^{\prime }+A_{hg}^{17}{T}_{17}^{\prime }+A_{hg}^{18}{T}_{18}^{\prime }\end{array}\right)}}\end{array}\mathrm{on} \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(8)

The notation used in Equation (8) is detailed in

Table 2. Notations used in Equation (8).

Index	Formula
${\mathrm{T}}_1^{\prime }$	0
${\mathrm{T}}_2^{\prime }$	0
${\mathrm{T}}_3^{\prime }$	$G{r}^{G-1}\left[\left(\cos (G\theta )\cos \theta +\sin (G\theta )\sin \theta \right)n_x+\left(\cos (G\theta )\sin \theta -\sin (G\theta )\cos \theta \right)n_y\right]$
${\mathrm{T}}_4^{\prime }$	$G{r}^{G-1}\left[\left(\sin (G\theta )\cos \theta -\cos (G\theta )\sin \theta \right)n_x+\left(\sin (G\theta )\sin \theta +\cos (G\theta )\cos \theta \right)n_y\right]$
${\mathrm{T}}_5^{\prime }$	$tG{r}^{G-1}\left[\left(\cos (G\theta )\cos \theta +\sin (G\theta )\sin \theta \right)n_x+\left(\cos (G\theta )\sin \theta -\sin (G\theta )\cos \theta \right)n_y\right]$
${\mathrm{T}}_6^{\prime }$	$tG{r}^{G-1}\left[\left(\sin (G\theta )\cos \theta -\cos (G\theta )\sin \theta \right)n_x+\left(\sin (G\theta )\sin \theta +\cos (G\theta )\cos \theta \right)n_y\right]$
${\mathrm{T}}_7^{\prime }$	$H{e}^{Hat}{I}_1(Hr)(\cos (\theta )n_x+\sin (\theta )n_y)$
${\mathrm{T}}_8^{\prime }$	$H{e}^{-Hat}{I}_1(Hr)(\cos (\theta )n_x+\sin (\theta )n_y)$
${\mathrm{T}}_9^{\prime }$	$-H\cos (Hat)J_1\left(Hr\right)\left[\cos (\theta )n_x+\sin (\theta )n_y\right]$
${\mathrm{T}}_{10}^{\prime }$	$-H\sin (Hat)J_1\left(Hr\right)\left[\cos (\theta )n_x+\sin (\theta )n_y\right]$
${\mathrm{T}}_{11}^{\prime }$	$H{e}^{Hat}\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\cos (G\theta )\cos \theta +I_G(Hr)\sin (G\theta ){\displaystyle \frac{\sin \theta }{r}}\right)n_x+\\ \left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\cos (G\theta )\sin \theta -I_G(Hr)\sin (G\theta ){\displaystyle \frac{\cos \theta }{r}}\right)n_y\end{array}\right]$
${\mathrm{T}}_{12}^{\prime }$	$H{e}^{Hat}\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\sin (G\theta )\cos \theta -I_G(Hr)\cos (G\theta ){\displaystyle \frac{\sin \theta }{r}}\right)n_x+\\ \left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\sin (G\theta )\sin \theta +I_G(Hr)\cos (G\theta ){\displaystyle \frac{\cos \theta }{r}}\right)n_y\end{array}\right]$
${\mathrm{T}}_{13}^{\prime }$	$H{e}^{-Hat}\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\cos (G\theta )\cos \theta +I_G(Hr)\sin (G\theta ){\displaystyle \frac{\sin \theta }{r}}\right)n_x+\\ \left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{q-1}(Hr))\cos (G\theta )\sin \theta -I_G(Hr)\sin (G\theta ){\displaystyle \frac{\cos \theta }{r}}\right)n_y\end{array}\right]$
${\mathrm{T}}_{14}^{\prime }$	$H{e}^{-Hat}\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\sin (G\theta )\cos \theta -I_G(Hr)\cos (G\theta ){\displaystyle \frac{\sin \theta }{r}}\right)n_x+\\ \left({\displaystyle \frac{1}{2}}(I_{G+1}(Hr)+I_{G-1}(Hr))\sin (G\theta )\sin \theta +I_G(Hr)\cos (G\theta ){\displaystyle \frac{\cos \theta }{r}}\right)n_y\end{array}\right]$
${\mathrm{T}}_{15}^{\prime }$	$H\cos (Hat)\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(J_{G-1}(Hr)-J_{G+1}(Hr))\cos (G\theta )\cos \theta +J_G(Hr)\sin (G\theta ){\displaystyle \frac{\sin \theta }{r}}\right)n_x+\\ \left({\displaystyle \frac{1}{2}}(J_{G+1}(Hr)-J_{G-1}(Hr))\cos (G\theta )\sin \theta -J_G(Hr)\sin (G\theta ){\displaystyle \frac{\cos \theta }{r}}\right)n_y\end{array}\right]$
${\mathrm{T}}_{16}^{\prime }$	$H\cos (Hat)\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(J_{G-1}(Hr)-J_{G+1}(Hr))\sin (G\theta )\cos \theta -J_G(Hr)\cos (G\theta )\sin \theta \right)n_x+\\ \left({\displaystyle \frac{1}{2}}(J_{G+1}(Hr)-J_{G-1}(Hr))\sin (G\theta )\sin \theta +J_G(Hr)\cos (G\theta )\cos \theta \right)n_y\end{array}\right]$
${\mathrm{T}}_{17}^{\prime }$	$H\sin (Hat)\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(J_{G-1}(Hr)-J_{G+1}(Hr))\cos (G\theta )\cos \theta +J_G(Hr)\sin (G\theta )\sin \theta \right)n_x+\\ \left({\displaystyle \frac{1}{2}}(J_{G+1}(Hr)-J_{G-1}(Hr))\cos (G\theta )\sin \theta -J_G(Hr)\sin (G\theta )\cos \theta \right)n_y\end{array}\right]$
${\mathrm{T}}_{18}^{\prime }$	$H\sin (Hat)\left[\begin{array}{l}\left({\displaystyle \frac{1}{2}}(J_{G-1}(Hr)-J_{G+1}(Hr))\sin (G\theta )\cos \theta -J_G(Hr)\cos (G\theta )\sin \theta \right)n_x+\\ \left({\displaystyle \frac{1}{2}}(J_{G+1}(Hr)-J_{G-1}(Hr))\sin (G\theta )\sin \theta +J_G(Hr)\cos (G\theta )\cos \theta \right)n_y\end{array}\right]$

. $\mathsf{\partial }{\mathsf{\Omega }}_t={\mathsf{\Gamma }}^D\cup {\mathsf{\Gamma }}^N$ is the whole boundary of the computational domain, and $n=(n_{\mathrm{x}},n_y)$ denotes the unit outward normal vector along the boundary. To determine the unknown coefficients of $A^1,A^2,A_g^3,\cdots ,A_{hg}^{18}$ in Equations (7) and (8), the ST collocation scheme is employed, resulting in the following matrices:

$$\left[\begin{array}{ccccc}1& t_1& r_1^g\cos (g{\theta }_1)& \cdots & \sin (ha{t}_1)\sin (g{\theta }_1)J_{{\lambda }_g}(h{r}_1)\\ 1& t_2& r_2^g\cos (g{\theta }_2)& \cdots & \sin (ha{t}_2)\sin (g{\theta }_2)J_{{\lambda }_g}(h{r}_2)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& t_v& r_v^g\cos (g{\theta }_v)& \cdots & \sin (ha{t}_v)\sin (g{\theta }_v)J_{{\lambda }_g}(h{r}_v)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& {T_3^{\prime }}_{(v)}& \cdots & {T_{18}^{\prime }}_{(v)}\end{array}\right]\left[\begin{array}{c}{A}^1\\ A^2\\ A_g^3\\ ⋮\\ A_{hg}^{18}\end{array}\right]=\left[\begin{array}{c}{u}_1\\ u_2\\ ⋮\\ u_v\\ ⋮\\ {\displaystyle \frac{\mathsf{\partial }{u}_v}{\mathsf{\partial }n}}\end{array}\right],$$

(9)

where $v$ represents the total number of boundary points. $H=h, G=g$ are obtained from the periodic condition using the Sommerfeld–Watson transformation [19]. Therefore, Equation (9) is obtained as follows:

$$\mathsf{\psi }A=U,$$

(10)

where $\mathsf{\psi }$ represents assembled from a matrix of the Trefftz basis functions, $A$ represents a vector of unknown coefficients, and $U$ denotes a vector of accessible boundary value at boundary points. By solving Equation (10), we obtain the unknown coefficients for the ST domain.

2.3. The Multiple-Scale Trefftz Method

The precision of computational solutions derived through SCTM is highly dependent on the placement of collocation points that enforce boundary constraints and the number of Trefftz trial functions employed. Given the inherently ill-conditioned nature of the linear equation system generated by SCTM, the numerical solution may exhibit instability. Thus, it is crucial to explore ways to improve the conditioning of this system. According to Ku [20], utilizing MSCL can mitigate the conditioning of an ill-conditioned linear equation system. To address wave problems, we implemented MSCL to prevent the occurrence of a severely ill-conditioned system. The MSCL $l_i$ is derived from matrix $\psi$ in Equation (10) through the following expression:

$$l_i=\sqrt{{\displaystyle \sum _{j=1}^v{\psi }_{j,i}^2}} , i=1,2,\cdots ,o,$$

(11)

where $o=2+4(m+s)+8ms$ is the number of Trefftz trial functions. The MSCL $l_i$ was added to Equation (9), which can be rewritten as follows:

$$\left[\begin{array}{ccccc}{\displaystyle \frac{1}{l_1}}& {\displaystyle \frac{t_1}{l_2}}& {\displaystyle \frac{r_1^g\cos (g{\theta }_1)}{l_3}}& \cdots & {\displaystyle \frac{\sin (ha{t}_1)\sin (g{\theta }_1)J_{{\lambda }_g}(h{r}_1)}{l_o}}\\ {\displaystyle \frac{1}{l_1}}& {\displaystyle \frac{t_2}{l_2}}& {\displaystyle \frac{r_2^g\cos (g{\theta }_2)}{l_3}}& \cdots & {\displaystyle \frac{\sin (ha{t}_2)\sin (g{\theta }_2)J_{{\lambda }_g}(h{r}_2)}{l_o}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{l_1}}& {\displaystyle \frac{t_v}{l_2}}& {\displaystyle \frac{r_v^g\cos (g{\theta }_v)}{l_3}}& \cdots & {\displaystyle \frac{\sin (ha{t}_v)\sin (g{\theta }_v)J_{{\lambda }_g}(h{r}_v)}{l_o}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& {\displaystyle \frac{{T_3^{\prime }}_{(v)}}{l_3}}& \cdots & {\displaystyle \frac{{T_{18}^{\prime }}_{(v)}}{l_o}}\end{array}\right]\left[\begin{array}{c}{\widehat{A}}^1\\ {\widehat{A}}^2\\ {\widehat{A}}_g^3\\ ⋮\\ {\widehat{A}}_{hg}^{18}\end{array}\right]=\left[\begin{array}{c}{u}_1\\ u_2\\ ⋮\\ u_v\\ ⋮\\ {\displaystyle \frac{\mathsf{\partial }{u}_v}{\mathsf{\partial }n}}\end{array}\right].$$

(12)

Applying the scales $l_i$ in Equation (12) considerably improves the conditioning of the linear equation system. In the multiple-scale Trefftz method, rather than solving for ${\left[\begin{array}{ccccc}{A}^1& A^2& A_{\mathrm{g}}^3& \cdots & A_{hg}^{18}\end{array}\right]}^{\mathrm{T}}$ in Equation (9), the unknown coefficients ${\left[\begin{array}{ccccc}{\widehat{A}}^1& {\widehat{A}}^2& {\widehat{A}}_g^3& \cdots & {\widehat{A}}_{hg}^{18}\end{array}\right]}^{\mathrm{T}}$ are determined. We can rewrite Equation (12) as follows:

$$\widehat{\mathsf{\psi }}\widehat{A}=U.$$

(13)

By adopting the multiple-scale method in this study, we achieved a substantial improvement in the conditioning of the system. This reduction allows us to use matrix left division to solve the system of simultaneous linear equations. The architecture illustrated in Figure 2 integrates a unified Multiple-Scale STCM, which combines Trefftz basis functions, MSCL scaling, and direct space–time collocation into a consistent one-step mesh-free strategy.

3. Accuracy Analysis

This section reports on the analysis of the convergence of the proposed method. All computations integrating SCTM with MSCL techniques were performed in MATLAB2022b. The error measures are defined as follows to quantify the accuracy of the approximations:

$$\mathrm{MAE}=\max \left|u_A(r_i,{\theta }_i,t_i)-u_N(r_i,{\theta }_i,t_i)\right|, 1\le i\le N$$

(14)

$$E_{L^2}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{(u_A(r_i, {\theta }_i, t_i)-u_N(r_i, {\theta }_i, t_i))}^2}}{{\displaystyle \sum _{i=1}^N{u}_A{(r_i, {\theta }_i, t_i)}^2}}}}$$

(15)

where $u_A(r_i,{\theta }_i,t_i)$ and $u_N(r_i,{\theta }_i,t_i)$ denote the analytical and numerical solutions, respectively, evaluated at the i-th validation point. MAE represents the maximum absolute error, $E_{L^2}$ denotes the $L^2$-norm error, and $N$ is the number of resolved points. An analysis of a 2D wave problem is conducted. The irregular geometry is defined as follows:

$${\mathsf{\Omega }}_t=\left\{(r,\theta ,t)\left|\begin{array}{l}r={\displaystyle \frac{3+\cos (\theta -1.5\pi )\sin (2\theta )}{3+\sin (10\theta )}},\\ 0\le \theta \le 2\pi , 0\le t\le 1\end{array}\right.\right\},$$

(16)

The initial data is the following:

$$u=0,(r,\theta ,t=0)\in {\mathsf{\Omega }}_t$$

(17)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}=\sqrt{2}\cos (r\cos \theta \left)\cos \right(r\sin \theta ),(r,\theta ,t=0)\in {\mathsf{\Omega }}_t$$

(18)

The boundary conditions are defined as follows:

$$u\left(r,\theta ,t\right)=\cos (r\cos \theta \left)\cos \right(r\sin \theta )\sin (\sqrt{2}t),(r,\theta ,t)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(19)

The precise solution is the following:

$$u\left(r,\theta ,t\right)=\cos (r\cos \theta \left)\cos \right(r\sin \theta )\sin (\sqrt{2}t).$$

(20)

In this example, the total elapsed time is 1, with a wave speed of 1. A convergence analysis is first conducted to validate our approach. As illustrated in Figure 3, there are boundary nodes and source nodes. The accuracy of our solution is examined by considering the order of Trefftz basis functions and the placement of boundary nodes.

To examine the orders s and m on the solution accuracy, the boundary points are set at 1355. Figure 4 depicts the error varies with changes in the order of the basis functions. High accuracy approximations seem to be achieved with m in the range of 12–30 and s in the range of 1–10. Figure 4 shows that as s increases from 1 to 5, the MAE decreases and as s increases further from 5 to 30, the MAE increases. When s = 5 the MAE as a function of m is shown in Figure 5, where it appears that the numerical solutions of the proposed SCTM have high accuracy at m = 12. The error is maintained at around ${10}^{-8}$ when m is greater than 12.

To analyze the influence of boundary points on solution accuracy, the values of s and m are chosen to be 5 and 12, respectively. Figure 6 illustrates how the error magnitude varies with the number of boundary points. The results showcase that precise approximations are achieved once the boundary point count surpasses 1000. Drawing from the findings of the stability study, the number of boundary nodes, as well as the values of s and $m$ are set to be 1035, 5, and 12, respectively. Under these settings, our method achieves MAE and $E_{L^2}$ in the order of ${10}^{-8}$ and ${10}^{-11}$, respectively.

4. Numerical Examples

4.1. Complicated Wave Vibrating Problem

Evaluating the precision of outcomes obtained through this method, in comparison with other numerical techniques, involves analyzing a more complex wave vibration problem featuring a smooth boundary and a specified wave speed equal to the unit. The boundary condition is as follows:

$$u(r,\theta ,t)=3+\cos ({\displaystyle \frac{\pi }{10}}r\cos \theta )\cos ({\displaystyle \frac{\pi }{10}}r\sin \theta )\sin {\displaystyle \frac{\sqrt{2}\pi t}{10}},(r,\theta ,t)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(21)

Equation (21) represents the precise solution of the 2D wave equation. Regarding the numerical example analyzed, two domains are considered, a square domain and an irregular domain. The profiles of the ST domains and boundary collocation points for these two shapes are illustrated in Figure 7. The profiles of the ST domains and boundary collocation points for these two shapes are illustrated in Figure 7. The defining equations for these domains are as follows:

$$\mathsf{\partial }{\mathsf{\Omega }}_1=\left\{(r,\theta , t)\left|\begin{array}{l}0\le \theta \le 2\pi ,0\le t\le 5,\\ r=\sqrt{x^2+y^2},0\le x\le 1,0\le y\le 1\end{array}\right.\right\},$$

(22)

$$\mathsf{\partial }{\mathsf{\Omega }}_2=\left\{(r,\theta ,t)\left|\begin{array}{l}r=10{\left(\cos (4\theta )+{\left({\displaystyle \frac{18}{5}}-{\sin }^2(4\theta )\right)}^{{\displaystyle \frac{1}{2}}}\right)}^{{\displaystyle \frac{1}{3}}},\\ 0\le \theta \le 2\pi ,0\le t\le 60\end{array}\right.\right\}.$$

(23)

The initial condition is specified as detailed below:

$$u=3,(r,\theta ,t=0)\in \mathsf{\partial }{\mathsf{\Omega }}_t$$

(24)

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}={\displaystyle \frac{\sqrt{2}\pi }{10}}\cos ({\displaystyle \frac{\pi }{10}}r\cos \theta )\cos ({\displaystyle \frac{\pi }{10}}r\sin \theta ),(r,\theta ,t=0)\in \mathsf{\partial }{\mathsf{\Omega }}_t$$

(25)

Within the square domain, there are 540 boundary points and 81 initial points, with the values of s and m are set to 6. The approximations obtained from the proposed method closely match the analytical solution, with an MAE of $4.29\times {10}^{-9}$. A comparison was made between the computational error associated with the proposed method and that of the multiple-direction Trefftz method [21]. According to Liu et al., the MAE of the multiple-direction Trefftz method is approximately on the order of ${10}^{-6}$.

In the irregular domain, 3050boundary points and 875 initial points are used, with $s=10$ and $m=12$. Figure 8 depicts the $E_{L^2}$ and MAE for the developed approach as the time progresses; the $E_{L^2}$ and MAEs remain below $2.44\times {10}^{-5}$.

Table 3. Comparison regarding the complicated wave vibrating problem in irregular domain.

Model	FEM [22]	MPS-MFS [22]	This Study
Points	15,981	1209	3925
$t = 12$	$1.18 \times {10}^{-4}$	$2.81 \times {10}^{-4}$	$2.07 \times {10}^{-6}$
$t = 24$	$3.70 \times {10}^{-5}$	$2.48 \times {10}^{-5}$	$2.70 \times {10}^{-6}$
$t = 36$	$2.67 \times {10}^{-4}$	$3.97 \times {10}^{-4}$	$1.61 \times {10}^{-6}$
$t = 48$	$5.27 \times {10}^{-4}$	$6.98 \times {10}^{-4}$	$7.90 \times {10}^{-7}$
$t = 60$	$4.07 \times {10}^{-4}$	$5.90 \times {10}^{-4}$	$2.87 \times {10}^{-6}$

demonstrates a comparison of the computational $E_{L^2}$ between the proposed approach and the FEM and MPS-MFS [22]. FEM and MPS-MFS have $E_{L^2}$ values of ${10}^{-4}$, while that of the proposed approach reaches ${10}^{-6}$. The solution maintains a high level of accuracy, even over extended time periods and within irregular domains. These simulations were executed within a CPU time of 6.2455 s. Compared to FEM, which shows a significant increase in error as time progresses, and MPS-MFS, which exhibits relatively consistent stability, the proposed method demonstrates exceptional stability. The errors remain low at every single time node, indicating superior numerical stability and robustness, even during longer simulations. Additionally, the proposed method uses fewer points (3925) while yielding similar or better accuracy, offering an enhanced computational efficiency by balancing precision and resource requirements.

4.2. Inverse Problem of Vibrations

In this study, we consider the case of the free vibrations of irregular membranes described by the following mathematical model. The inverse problem of these vibrations is represented by Equation (2) and a wave speed of 1. An illustrative diagram of the computational region and boundary collocation points is shown in Figure 9, defined as follows:

$$\mathsf{\partial }{\mathsf{\Omega }}_t=\left\{(r,\theta ,t)\left|\begin{array}{l}r=0.4(3+{\cos }^{16}(2\theta +0.5\pi )),\\ 0\le \theta \le 2\pi ,0\le t\le 2\end{array}\right.\right\}.$$

(26)

The boundary conditions are defined as follows:

$$u(r,\theta ,t)=\sin (0.5\pi (r\cos \theta +1))\sin (0.5\pi (r\sin \theta +1))\cos (\sqrt{0.5}\pi t),(r,\theta ,t)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(27)

Moreover, the wave field at t = 2 is defined as follows:

$$u(r,\theta ,t)=\sin (0.5\pi (r\cos \theta +1))\sin (0.5\pi (r\sin \theta +1))\cos (2\sqrt{0.5}\pi ),(r,\theta ,t)\in {\mathsf{\Omega }}_t.$$

(28)

The velocity distribution of the wave field at t = 2 is computed as follows:

$${\displaystyle \frac{\mathsf{\partial }u(r,\theta ,2)}{\mathsf{\partial }t}}=-\sqrt{0.5}\pi \sin (0.5\pi (r\cos \theta +1))\sin (0.5\pi (r\sin \theta +1))\sin (2\sqrt{0.5}\pi ),(r,\theta ,t=2)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(29)

Finally, the analytical solution is determined to be the following:

$$u(r,\theta ,t)=\sin (0.5\pi (r\cos \theta +1))\sin (0.5\pi (r\sin \theta +1))\cos (\sqrt{0.5}\pi t),(r,\theta ,t)\in {\mathsf{\Omega }}_t.$$

(30)

In this instance, the values for s and m are set to 6 and 20, in that order. There are 960 boundary points and 1367 final time condition points. The SCTM scheme is presented in Figure 9 with 2327 boundary points, whose MAE and $E_{L^2}$ are $2.05\times {10}^{-9}$ and $3.39\times {10}^{-10}$, respectively. Figure 10 presents the numerical and analytical solutions obtained using the proposed methods.

In this case, the accessible data contaminated by different random noises are considered.

$${\tilde{F}}_D(r, \theta , t)=F_D(r, \theta , t)\times \left[1+{\displaystyle \frac{ns}{100}}\times (2\times rand-1)\right],$$

(31)

$${\displaystyle \frac{\mathsf{\partial }\tilde{u}(r, \theta , t)}{\mathsf{\partial }t}}={\displaystyle \frac{\mathsf{\partial }u(r, \theta , t)}{\mathsf{\partial }t}}\times \left[1+{\displaystyle \frac{ns}{100}}\times (2\times rand-1)\right],$$

(32)

where $ns$ denotes the percentage of noise, ${\tilde{F}}_D$ highlights the noisy Dirichlet data, ${\displaystyle \frac{\mathsf{\partial }\tilde{u}(r,\theta ,t)}{\mathsf{\partial }t}}$ represents the noisy velocity distribution of the wave field, and rand is a random number.

The boundary data, which is contaminated by noise, is further analyzed with varying noise levels, specifically, ns = 0, 0.001, 0.01, 0.1, and 1.

Table 4. Error analysis of the proposed approach for different noise levels.

Noise Level	$\mathit{n}\mathit{s}=0$	$\mathit{n}\mathit{s}=0.001$	$\mathit{n}\mathit{s}=0.01$	$\mathit{n}\mathit{s}=0.1$	$\mathit{n}\mathit{s}=1$
MAE	$2.05\times {10}^{-9}$	$8.25\times {10}^{-5}$	$5.41\times {10}^{-3}$	$9.52\times {10}^{-2}$	$5.70\times {10}^{-1}$
$E_{L^2}$	$3.39\times {10}^{-10}$	$7.34\times {10}^{-4}$	$8.85\times {10}^{-4}$	$1.6\times {10}^{-2}$	$7.47\times {10}^{-2}$

presents the error values for each of these noise levels. It is observed that the proposed method demonstrates a certain level of noise resistance, as the errors in MAE and $E_{L^2}$ remain within acceptable limits $5.70\times {10}^{-1}$ and $7.47\times {10}^{-2}$, respectively, even when the noise level reaches ns = 1.

4.3. Mixed-Boundary Condition Problem

In this case, the 2D wave problem under mixed-boundary conditions is considered. The schematic representation of the simulation area and boundary collocation points is shown in Figure 11, defined as follows:

$$\mathsf{\partial }{\mathsf{\Omega }}_t=\left\{(r,\theta ,t)\left|\begin{array}{l}r=\sqrt[3]{\cos (3\theta )+\sqrt{2-{\sin }^2(3\theta )}},\\ 0\le \theta \le 2\pi , 0\le t\le 2\end{array}\right.\right\},$$

(33)

The governing equation is Equation (2) with a wave speed of $5$. Dirichlet boundary conditions can be defined as follows:

$$u(r,\theta ,t)=J_0(r)\left[\sin (5t)+\cos (5t)\right],(r,\pi \le \theta <2\pi ,t)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(34)

In addition, the Neumann boundary conditions are given by:

$$u_n={\displaystyle \frac{\mathsf{\partial }u(r,\theta ,t)}{\mathsf{\partial }n}},(r,0\le \theta <\pi ,t)\in \mathsf{\partial }{\mathsf{\Omega }}_t.$$

(35)

The initial condition is specified as follows:

$$u(r,\theta ,0)=J_0(r),(r,\theta ,t=0)\in \mathsf{\partial }{\mathsf{\Omega }}_t$$

(36)

$${\displaystyle \frac{\mathsf{\partial }u(r,\theta ,0)}{\mathsf{\partial }t}}=5J_0(r)\cos (5t),(r,\theta ,t=0)\in \mathsf{\partial }{\mathsf{\Omega }}_t$$

(37)

Finally, the analytical solution is found to be:

$$u(r,\theta ,t)=J_0(r)\left[\sin (5t)+\cos (5t)\right],(r,\theta ,t)\in {\mathsf{\Omega }}_t.$$

(38)

In this case, the values of s and m are 5 and 12, respectively. The proposed scheme is presented in Figure 11, with 5000 Neumann boundary points, 5000 Dirichlet boundary points, 218 initial conditions. Under mixed-boundary conditions, the MAE between the analytical solution and the numerical solution in this case can reach approximately ${10}^{-12}$. Figure 12 illustrates the absolute error between the solution and numerical solutions at different time instances.

5. Conclusions

In this study, we developed a one-step, semi-analytical mesh-free SCTM for 2D wave equations. The method’s originality lies in a unified ST Trefftz basis that treats time as an analytic variable, removing time marching and limiting truncation-error accumulation, together with an MSCL scaling that regularizes the collocation system and improves numerical stability. Across benchmarks and applications, the approach accurately reconstructs wave fields in complex geometries and under mixed boundary conditions. Accuracy analysis shows that our method achieves MAE and $E_{L^2}$ on the order of ${10}^{-8}$ and ${10}^{-11}$, respectively, while also demonstrating robustness toward noise. The framework preserves the geometric flexibility of mesh-free discretization while providing a direct ST treatment, making it well suited for long-duration simulations and backward analyses. Nevertheless, the present study is limited to 2D cases, and future work will extend SCTM to three-dimensional and nonlinear problems to broaden its applicability.