Mathematics and Computer Science, Volume 1

18Numerical Solution of Partial Differential Equations: Finite Difference Method

Roushan Kumar¹, Rakhi Tiwari²* and Rashmi Prasad¹

¹Department of Mathematics, Central University of South Bihar, Gaya, Bihar, India

²Department of Mathematics, Nitishwar College, a Constituent Unit of Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, Bihar, India

Abstract

This book chapter provides a broad description of the finite difference methods for parabolic differential equations (heat equation). Section 18.1 covers an overview of second-order partial differential equation via: classification, initial, and boundary conditions. Section 18.2 discusses the finite difference method, in which we provide the discretization of the domain and finite difference approximation of heat equation and some primary definitions (consistency, convergence, stability). Section 18.3 describes the general explicit method and its characteristics, e.g., truncation error, stability (non-Neumann), and some well-known explicit methods. Section 18.4 presents general two-level implicit methods and some characteristics, e.g., truncation error, stability (non-Neumann), an explicit method. Lastly, we provide the numerical example solved by a different numerical method for verification and validation of the computed solution; after that, concluding remarks are presented.

Keywords: Partial differential equations, finite difference method

18.1 Introduction

Partial differential equations (PDE) arise in the mathematical ...