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Chapter 3 - Classification of Quasi-Linear Partial Differential Equations

Published online by Cambridge University Press:  05 January 2014

Tapan Sengupta
Affiliation:
Indian Institute of Technology, Kanpur
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Summary

Introduction

In Chapter 2, we have derived most general form of conservation laws which are solved in CFD. For continuum flows, these are given by Navier-Stokes equation. A quick look at them will show that these are non-linear partial differential equations. To be more precise, they are linear with respect to the highest derivative terms and such equations are called quasi-linear PDEs. It is possible to classify such equations based on the behavior of their solutions. In this chapter, we classify quasilinear PDEs, so that we can derive specific numerical methods for each equation Type in subsequent chapters. Despite the observation that different numerical methods are chosen for different classes of PDEs, we will also see here and in later chapters, that there is a generality of approach in treating these PDEs, whose solution and error propagate in time in a unified manner. This will be clearly evident even for time-independent problems, which are solved iteratively, as will be demonstrated through an example in Section 3.3.

Classification of Partial Differential Equations

Consider the moving boundary problem as shown below. This class of problems is also known as the propagation problem.

Type
Chapter
Information
High Accuracy Computing Methods
Fluid Flows and Wave Phenomena
, pp. 31 - 37
Publisher: Cambridge University Press
Print publication year: 2013

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