Numerical Methods for Hyperbolic PDEs

Numerical methods for hyperbolic PDEs like the wave equations, Maxwell’s equations of electromagnetism, the inviscid Burgers equation, gas dynamics, and magnetohydrodynamics are completely different from numerical methods for parabolic PDEs. In hyperbolic PDEs, information propagates along characteristic curves in the form of waves with finite velocity. Mathematically appropriate boundary conditions for hyperbolic PDEs are Cauchy, which are based on characteristics. First-order hyperbolic methods (like the first-order upwind and Lax-Friedrichs methods) are diffusive since they spread out propagating waves through numerical diffusion. Second-order hyperbolic methods (like Lax-Wendroff) are dispersive since they introduce dispersive oscillations in propagating waves. Upwind methods are preferred, especially modern high-order upwind methods that minimize the amount of numerical diffusion and dispersion. Numerical methods for hyperbolic conservation laws should conserve quantities that are conserved for the original continuous PDEs, so conservative numerical methods are always advocated. The methods of choice for nonlinear hyperbolic problems like gas dynamics are WENO or higher-order Godunov methods like PPM and CLAWPACK. Here we focus on WENO, but Godunov’s method is also presented and discussed. WENO3 simulations of Riemann problems for Burgers’ equation and gas dynamics are presented, as well as 2D WENO3 simulations of supersonic jets.