Finite difference methods for PDEs

This chapter introduces the fundamentals of numerical solution of partial differential equations (PDEs), starting with the upwind scheme for first-order PDEs, followed by von Neumann stability analysis. Numerical methods such as the central difference scheme are then explained for second-order PDEs such as wave equation, heat conduction equation, and Poisson's equation, together with stability analysis of these schemes. Finally, the basic idea of the spectral method is outlined.