Classical Finite Difference Methods for Hyperbolic and Discontinuities

This chapter introduces classical finite difference methods for solving hyperbolic partial differential equations, with emphasis on resolving discontinuities and steep gradients encountered in compressible flow and gas dynamics. The chapter begins with the method of characteristics, demonstrating how nonlinear wave steepening and shock formation arise from intersecting characteristics. It then presents shock-capturing techniques such as Lax-Friedrichs, Lax-Wendroff, MacCormack, Beam–Warming, and CIR schemes, analyzing their stability, accuracy, and numerical dispersion through amplification factor derivations and Fourier analysis. The CFL condition is discussed in the context of stability. The chapter addresses the Gibbs phenomenon, its origin in high-order schemes near discontinuities, and the trade-off between numerical diffusion and resolution. The case of the inviscid Burgers&s; equation is explored.