The swimpde framework presents an efficient, mesh-free alternative to traditional solvers for Partial Differential Equations (PDEs) using the ”Sampling Where It Matters” (SWIM) framework. The motivation for this thesis stems from the need to determine where we actually require the basis functions for a given simulation. This thesis extends the capa- bilities of the swimpde framework to robustly solve the 1D and 2D wave equations with adaptive basis functions. The primary contributions are threefold. First, we identify and correct the multi-block resampling strategy for second-order time-dependent PDEs, ensur- ing that the full physical state, including time derivatives (u and ut ), is correctly transferred between blocks. Second, we implement an adaptive resampling method for 2D problems by implementing a flexible acceptance-rejection sampling algorithm, moving beyond the limitations of the 1D inverse transform method. Third, we develop a new, robust probabil- ity density function (PDF) for resampling. This new PDF incorporates a Gaussian filter to smooth the solution gradients, which has proven essential for stable tracking of oscillatory wavefronts and preventing error amplification upon reflection, a problem that plagued naive gradient-based approaches. These contributions are validated through a series of numerical experiments. The en- hanced solver accurately matches analytical solutions for 1D and 2D wave propagation problems, which include a 1D reflecting Gaussian pulse and a 2D droplet simulation. The results demonstrate that the new PDF is efficient in terms of numerical stability for reflec- tion problems, and the 2D resampling strategy effectively captures complex, propagating wave patterns for 2D wave problems.     «

The swimpde framework presents an efficient, mesh-free alternative to traditional solvers for Partial Differential Equations (PDEs) using the ”Sampling Where It Matters” (SWIM) framework. The motivation for this thesis stems from the need to determine where we actually require the basis functions for a given simulation. This thesis extends the capa- bilities of the swimpde framework to robustly solve the 1D and 2D wave equations with adaptive basis functions. The primary contributions are t...     »