Numerical Methods for Partial Differential Equations

Abstract

Partial Differential Equations (PDEs) are the mathematical engine of many models that have applications in the physical, biologic and engineering sciences. Numerical methods are essential to get approximate solutions if analytical solutions are not available. This article gives a full treatment of numerical methods for PDEs in the style of a research article with each section to address the theory, algorithmic choices, implementation approaches, and empirical comparisons. Finite difference, finite element, finite volume, spectral and discontinuous Galerkin methods are presented along with focuses on consistency, stability, convergence, adaptivity and solver performance. Part of the evaluation includes the use of representative model problems to assess accuracy, computational cost and robustness of models for several elliptic, parabolic and hyperbolic problems. The work ends with a set of practical recommendations, as well as a list of selected suggestions for genuine references for further study.