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Time-dependent Partial Differential Equations and Their Numerical Solution

  • 2001
  • Book
Authors
Heinz-Otto Kreiss
Hedwig Ulmer Busenhart
Book Series
Lectures in Mathematics ETH Zürich
Publisher
Birkhäuser Basel
Included in
Professional Book Archive
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About this book

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

Frontmatter
Chapter 1. Cauchy Problems
Abstract
We consider the initial value problem
$$ \begin{gathered} u_t (x,t) + u_x (x,t) = 0,x \in \mathbb{R},t \geqslant 0, \hfill \\ u(x,0) = f(x),x \in \mathbb{R}, \hfill \\ \end{gathered} $$
(1.1.1)
where f (x) = f (x + 2π) is a smooth 2π-periodic function. To begin, we assume that
$$ f(x) = e^{i\omega x} \hat f(\omega ) $$
(ω) denotes the Fourier transform of f (x). In order to construct a solution of the same type we choose the ansatz
$$ u(x,t) = e^{i\omega x} \hat u(\omega ,t). $$
(1.1.2)
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart
Chapter 2. Half Plane Problems
Abstract
In this section, we discuss the concept of well-posedness for half plane problems. Consider a strongly hyperbolic system
$$ \begin{gathered} u_t (x,t) = Au_x (x,t) + F(x,t) for x \geqslant 0, t \geqslant 0, \hfill \\ u(x,0) = f(x) for x \geqslant 0, \hfill \\ \end{gathered} $$
(2.1.1)
We assume that A is a constant nonsingular diagonal matrix and use the notation u = (u1…u n ) T . We write A in the form
$$ A = \left( {\begin{array}{*{20}c} { - \Lambda _1 } \hfill & 0 \hfill \\ 0 \hfill & {\Lambda _2 } \hfill \\ \end{array} } \right), $$
such that Λ1 = diag (λ1,…λ r ) > 0 and Λ2 = diag (λr+1,…λ n ) > 0. The initial data f (x) and the forcing fuction F (x, t) are smooth functions, which belong to L2, i.e.
$$ \left\| f \right\|_{L^2 }^2 = \int_0^\infty {\left| f \right|} ^2 dx < \infty ,\left\| {F( \cdot ,t)} \right\|_{L^2 }^2 < \infty , $$
for every fixed t. In addition, we give boundary conditions
$$ Lu(0,t) = g(t), $$
(2.1.2)
where L stands for q linearly independent relations
$$ L = \left( {\begin{array}{*{20}c} {l_{11} } \hfill & \cdots \hfill & {l_{1n} } \hfill \\ \vdots \hfill & {} \hfill & \vdots \hfill \\ {l_{q1} } \hfill & \cdots \hfill & {l_{qn} } \hfill \\ \end{array} } \right). $$
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart
Chapter 3. Difference Methods
Abstract
Consider the problem
$$ {{u}_{t}}(x,t) = {{u}_{x}}(x,t), - \infty \leqslant x < \infty ,t \geqslant 0, $$
(3.1.1)
with initial data
$$u(x,0) = f(x)$$
(3.1.2)
. We assume that f (x) and therefore also the solution of (3.1.1) is 2π-periodic. We discretize Equation (3.1.1) in space but keep time continuous.
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart
Chapter 4. Nonlinear Problems
Abstract
In this section we shall discuss nonlinear problems. We are interested in smooth solutions. There is no general theory for nonlinear differential equations available. Instead, we ask the following questions. Assume that we know a solution U for a particular set of data. Is the problem still solvable if we make small perturbations of the data? Does the solution depend continuously on the perturbation, i.e., do small perturbations in the data generate small changes in the solution?
Heinz-Otto Kreiss, Hedwig Ulmer Busenhart
Backmatter
Metadata
Title
Time-dependent Partial Differential Equations and Their Numerical Solution
Authors
Heinz-Otto Kreiss
Hedwig Ulmer Busenhart
Copyright Year
2001
Publisher
Birkhäuser Basel
Electronic ISBN
978-3-0348-8229-3
Print ISBN
978-3-7643-6125-9
DOI
https://doi.org/10.1007/978-3-0348-8229-3