The Accurate Finite-difference Scheme and Finite-element Method for some Partial Differential Equations

This monograph focus is targeted to graduate students, researchers and lecturers. It is devoted to construction of the high-order schemes of finite difference method and the finite element method for solving the multidimensional boundary value problems for different partial differential equations, in particular, linear Helmholtz and wave equations, and nonlinear Burgers' equation. The finite-difference method is a standard numerical method for solving boundary value problems. Recently, considerable attention has been attracted to construct a accurate (or exact) difference approximation for some ordinary and partial differential equations [1, 2, 3]. A accurate finite-difference method is developed for Helmholtz and wave equations with general boundary conditions (including initial condition for wave equation) on the rectangular domain in R2. The method proposed here comes out from [4] and is based on separation of variables method and expansion of one-dimensional three point difference operators for sufficiently smooth solution. The efficiency and accuracy of method were tested on several examples.