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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.
Details
Finite volume method;
Accuracy;
Partial differential equations;
Inverse problems;
Time marching;
Collocation methods;
Linear systems;
Numerical analysis;
Wave propagation;
Finite element analysis;
Trefftz method;
Systems stability;
Feasibility;
Wave equations;
Boundary conditions;
Collocation;
Linear equations;
Boundary value problems
; Cheng-Yu, Ku 4
; He, Xi 5 ; Chang-Kai, Lu 5 1 School of Smart Marine Science and Technology, Fujian University of Technology, Fuzhou 350118, China, Key Laboratory of Marine Environmental Survey Technology and Application, Ministry of Natural Resources, Guangzhou 510300, China, Fujian Provincial Key Laboratory of Marine Smart Equipment, Fujian University of Technology, Fuzhou 350118, China
2 School of Computer Science and Mathematics, Fujian University of Technology, Fuzhou 350118, China
3 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, [email protected] (C.-Y.K.), Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
4 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, [email protected] (C.-Y.K.)
5 Key Laboratory of Marine Environmental Survey Technology and Application, Ministry of Natural Resources, Guangzhou 510300, China