ABSTRACT
Application of numerical method to model partial differential equations is the focus of this chapter. Applications of various explicit and implicit methods to the simple one-dimensional wave or advection equation include Euler, Lax and Lax–Wendroff, leap frog, MacCormack, upwind, Runge–Kutta, trapezoidal differencing, Rusanov, Warming–Kutler–Lomax, and the discontinuous Galerkin scheme. Convergence, stability, and accuracy are discussed for each scheme and examples are included. Numerous techniques for computing solutions to the model parabolic equation, the heat equation, are discussed in detail. These include both explicit and implicit schemes. The main schemes considered include simple explicit, Crank–Nicolson, DuFort–Frankel, Keller box, alternating direction explicit and implicit, hopscotch, and discontinuous Galerkin. Additional details on issues encountered in heat and mass transfer problems are discussed including mass transport, contact resistance, and conduction across a material interface. A number of direct and iterative numerical schemes for computing solutions of linear algebraic system resulting from discrete models of Laplace’s equation are presented. Cramer’s rule, Gaussian elimination, Gauss–Seidel iteration, successive overrelaxation, coloring and alternating direction implicit (ADI) schemes, Krylov subspace methods, and the multigrid method are included. Methods for computing numerical solutions to the Burgers’ equation as a nonlinear convection diffusion model equation are also considered. For the inviscid case, Lax–Wendroff, MacCormack, Godunov, Roe, and Engquist–Osher schemes are given and the idea of a total variation diminishing (TVD) method is introduced. The primary methods applied to the viscous Burgers’ equation are the forward-time, centered-space (FTCS), leap frog/DuFort–Frankel, MacCormack, Briley–McDonald, ADI, Roe and the predictor–corrector, multiple-iteration methods. Throughout the chapter, examples are given showing the details encountered in the application of these methods.
