Time-Dependent Partial Differential Equations and Their Numerical Solution

by Kreiss, Heinz-Otto; Busenhart, Hedwig Ulmer

• ISBN13: 9783764361259
• ISBN10: 3764361255
• Format: Paperback
• Copyright: 2001-05-01
• Publisher: Birkhauser

Summary

In these notes we study time-dependent partial differential equations and their numerical solution. The analytic and the numerical theory are developed in parallel. For example, we discuss well-posed linear and nonlinear problems, linear and nonlinear stability of difference approximations and error estimates. Special emphasis is given to boundary conditions and their discretization. We develop a rather general theory of admissible boundary conditions based on energy estimates or Laplace transform techniques. These results are fundamental for the mathematical and numerical treatment of large classes of applications like Newtonian and non-Newtonian flows, two-phase flows and geophysical problems.

Table of Contents

Preface	vii
Cauchy Problems
Introductory Examples	1	(2)
Well-Posedness	3	(2)
Hyperbolic Systems with Constant Coefficients	5	(6)
In One Space Dimension	5	(2)
Symmetrizer	7	(1)
Multiple Space Dimensions	8	(3)
General Systems with Constant Coefficients	11	(8)
Parabolic Systems	14	(2)
Mixed Systems	16	(3)
Linear Systems with Variable Coefficients	19	(1)
Remarks	20	(1)
Half Plane Problems
Hyperbolic Systems in One Dimension	21	(4)
Hyperbolic Systems in Two Dimensions	25	(7)
Well-Posed Half Plane Problems	32	(2)
Well-Posed Problems in the Generalized Sense	34	(7)
Farfield Boundary Conditions	41	(3)
Energy Estimates	44	(1)
First Order Systems with Variable Coefficients	45	(1)
Remarks	46	(1)
Difference Methods
Periodic Problems	47	(9)
Half Plane Problems	56	(8)
Energy Estimates	57	(2)
Estimates by using Laplace Transform	59	(5)
Error Estimates	64	(1)
Method of Lines	64	(1)
Remarks	65	(2)
Nonlinear Problems
General Discussion	67	(1)
Initial Value Problems for Ordinary Differential Equations	67	(4)
Existence Theorems for Nonlinear Partial Differential Equations	71	(4)
Perturbation Expansion	75	(1)
Convergence of Difference Methods	75	(2)
Remarks	77	(2)
Bibliography	79	(2)
Index	81