1 - Partial Differential Equations

Summary

Introduction

Most physical as well as engineering systems one encounters in real life can be mathematically modeled using a system of partial differential equations subject to appropriate boundary conditions. These partial differential equations are coupled as well as nonlinear in nature. Owing to their nonlinearity, systems of partial differential equations that represent physical and engineering phenomena do not have closedform or analytical solutions. Thus, the only alternative available to a scientist or a engineer is to seek a numerical solution for the aforementioned systems of partial differential equations.

There are countless examples of the manifestation of partial differential equations with appropriate boundary conditions in various fields of physics, including magnetism, optics, statistical physics, general relativity, superconductivity, liquid crystals, turbulent flow in plasma and solitons. Furthermore, diverse fields such as fluid mechanics, atmospheric physics, and ocean physics have rich and exhaustive examples of partial differential equations. In this book an effort has been made to familiarize the readers to a general introduction of partial differential equations as well as equations of fluid motion before acquainting them with the various numerical methods. The well-known method of finite differences is introduced and important aspects such as consistency and stability are discussed while applying the above method to standard partial differential equations of the parabolic, hyperbolic, and elliptic types. The method of finite differences is then applied to equations of motion of the atmosphere and oceans. The book also introduces the readers to advanced numerical methods such as semi-Lagrangian methods, spectral method, finite volume, and finite element methods and provides for the application of the above methods to the equations of motion of the atmosphere and oceans.

Towards this end, it is important to introduce partial differential equations (PDE) and the various numerical methods that can be employed to solve PDEs numerically. A PDE is an equation that represents a relationship between an unknown function of two or more independent variables and the partial derivatives of this unknown function with respect to the independent variables. Although the independent variables are either space (x,y,z) or space and time (x,y,z,t) related, the nature of the unknown function depends on the physical/engineering problem being modeled.