Applications of compactly supported wavelets to the numerical solution of partial differential equations

摘要

Compactly supported wavelets have several properties that are useful for representing solutions of PDEs. The orthogonality, compact support and exact representation of polynomials of a fixed degree allow the efficient and stable calculation in regions with strong gradients or oscillations. The general method is a straightforward adaptation of the Galerkin procedure with a wavelet basis. Boundary conditions are imposed by a capacitance matrix method. Among the equations solved by these methods are the Burgers equation, the equations of Gas dynamics, the Euler and Navier-Stokes equations for an incompressible fluid in two dimensions with boundary conditions, the Schrodinger equation in two dimensions with singular particle potentials, the heat equation in two dimensions with boundary conditions and a discontinuous coefficient of diffusion, and the wave equation in two dimensions with a discontinuous sound speed (layered media). We present examples of the wavelet-Galerkin method applied to: the calculation of shocks for the Burgers equation, the calculation of the vortex dynamics for the Euler and Navier-Stokes equations, and calculations of solutions for the heat and wave equations in layered media.