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

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.