Discrete Wavelet Transform Projection Methods in Solving Partial Differential Equations With Multiscale Features

Accurate and efficient solutions of partial differential equations (PDEs) are central to engineering simulations, particularly for problems involving multiple physical processes and a wide range of spatial and temporal scales. These problems pose two main challenges: resolving fine-scale features without excessive computational cost, and handling stiff dynamics that can slow or destabilize solvers. Galerkin methods address these challenges by projecting the solution onto a set of basis functions, effectively reformulating the problem to separate scales. In this work, we focus on the Discrete Wavelet Transform (DWT) as a Galerkin basis. The DWT decomposes a function into localized components at multiple scales, providing compact support and orthogonality that facilitate accurate multiscale representation while favoring sparse representation. We demonstrate that DWT-based projection captures multiscale fluctuations with convergence as resolution increases. Compared with traditional finite-difference approaches, the method reduces numerical dissipation and improves stability while the solution remains smooth. Once discontinuities form, the method requires additional treatment to maintain stability. It also better preserves energy across scales. Remaining challenges, such as boundary treatment and discontinuity handling, are discussed, along with potential strategies to address them. These results indicate that wavelet-Galerkin formulations provide a practical foundation for developing numerical schemes capable of efficiently handling nonlinear, multiscale PDEs. By combining scale separation, sparsity, and energy preservation, this approach can accelerate computations while maintaining accuracy in simulations of complex engineering flows.