Time-dependent Partial Differential Equations and Their Numerical Solution

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.

1 Cauchy Problems.- 1.1 Introductory Examples.- 1.2 Well-Posedness.- 1.3 Hyperbolic Systems with Constant Coefficients.- 1.4 General Systems with Constant Coefficients.- 1.5 Linear Systems with Variable Coefficients.- 1.6 Remarks.- 2 Half Plane Problems.- 2.1 Hyperbolic Systems in One Dimension.- 2.2 Hyperbolic Systems in Two Dimensions.- 2.3 Well-Posed Half Plane Problems.- 2.4 Well-Posed Problems in the Generalized Sense.- 2.5 Farfield Boundary Conditions.- 2.6 Energy Estimates.- 2.7 First Order Systems with Variable Coefficients.- 2.8 Remarks.- 3 Difference Methods.- 3.1 Periodic Problems.- 3.2 Half Plane Problems.- 3.3 Method of Lines.- 3.4 Remarks.- 4 Nonlinear Problems.- 4.1 General Discussion.- 4.2 Initial Value Problems for Ordinary Differential Equations.- 4.3 Existence Theorems for Nonlinear Partial Differential Equations.- 4.4 Perturbation Expansion.- 4.5 Convergence of Difference Methods.- 4.6 Remarks.