ALBERT

Note: Zugl.: Braunschweig, Techn. Univ., Habil.-Schr., 2001

Note: Table of Contents: I BASIC CONCEPTS AND DISCRETIZATIONS. - 1 Advection-Diffusion-Reaction Equations. - 1.1 Nonlinear Reaction Problems from Chemistry. - 1.2 Model Equations for Advection-Diffusion. - 1.3 Multi-dimensional Problems. - 1.4 Examples of Applications. - 2 Basic Discretizations for ODEs. - 2.1 Initial Value Problems and Euler's Method. - 2.2 Norms and Matrices. - 2.3 Perturbations on ODE Systems. - 2.4 The θ-Method and Stiff Problems. - 2.5 Stability of the θ-Method. - 2.6 Consistency and Convergence of the θ-Method. - 2.7 Nonlinear Results for the θ-Method. - 2.8 Concluding Remarks. - 3 Basic Spatial Discretizations. - 3.1 Discrete Fourier Decompositions. - 3.2 The Advection Equation. - 3.3 The Diffusion Equation. - 3.4 The Advection-Diffusion Equation. - 4 Convergence of Spatial Discretizations. - 4.1 Stability, Consistency and Convergence. - 4.2 Advection-Diffusion with Constant Coefficients. - 4.3 Advection with Variable Coefficients. - 4.4 Diffusion with Variable Coefficients. - 4.5 Variable Coefficients and Higher-Order Schemes. - 5 Boundary Conditions and Spatial Accuracy. - 5.1 Refined Global Error Estimates. - 5.2 Outflow with Central Advection Discretization. - 5.3 Boundary Conditions with the Heat Equation. - 5.4 Boundary Conditions and Higher-Order Schemes. - 6 Time Stepping for PDEs. - 6.1 The Method of Lines and Direct Discretizations. - 6.2 Stability, Consistency and Convergence. - 6.3 Stability for MOL - Stability Regions. - 6.4 Von Neumann Stability Analysis. - 7 Monotonicity Properties. - 7.1 Positivity and Maximum Principle. - 7.2 Positive Semi-discrete Systems. - 7.3 Positive Time Stepping Methods. - 7.4 Numerical Illustrations. - 8 Numerical Test Examples. - 8.1 The Nonlinear Schrödinger Equation. - 8.2 The Angiogenesis Model. - II TIME INTEGRATION METHODS. - 1 Runge-Kutta Methods. - 1.1 The Order Conditions. - 1.2 Examples. - 1.3 The Stability Function. - 1.4 Step Size Restrictions for Advection-Diffusion. - 1.5 Rosenbrock Methods. - 2 Convergence of Runge-Kutta Methods. - 2.1 Order Reduction. - 2.2 Local Error Analysis. - 2.3 Global Error Analysis. - 2.4 Concluding Notes. - 3 Linear Multistep Methods. - 3.1 The Order Conditions. - 3.2 Examples. - 3.3 Stability Analysis. - 3.4 Step Size Restrictions for Advection-Diffusion. - 3.5 Convergence Analysis. - 4 Monotone ODE Methods. - 4.1 Linear Positivity for One-Step Methods. - 4.2 Nonlinear Positivity for One-Step Methods. - 4.3 Positivity for Multistep Methods. - 4.4 Related Monotonicity Results. - 5 Variable Step Size Control. - 5.1 Step Size Selection. - 5.2 An Explicit Runge-Kutta Example. - 5.3 An Implicit Multistep Example. - 5.4 General Purpose ODE Codes. - 6 Numerical Examples. - 6.1 A Model for Antibodies in Tumorous Tissue. - 6.2 The Nonlinear Schrödinger Equation. - III ADVECTION-DIFFUSION DISCRETIZATIONS. - 1 Non-oscillatory MOL Advection Discretizations. - 1.1 Spatial Discretization for Linear Advection. - 1.2 Numerical Examples. - 1.3 Positivity and the TVD Property. - 1.4 Nonlinear Scalar Conservation Laws. - 2 Direct Space-Time Advection Discretizations. - 2.1 Optimal-Order DST Schemes. - 2.2 A Non-oscillatory Third-Order DST Scheme. - 2.3 Explicit Schemes with Unconditional Stability. - 3 Implicit Spatial Discretizations. - 3.1 Order Conditions. - 3.2 Examples. - 3.3 Stability and Convergence. - 3.4 Monotonicity. - 3.5 Time Integration Aspects. - 4 Non-uniform Grids - Finite Volumes (1D). - 4.1 Vertex Centered Schemes. - 4.2 Cell Centered Schemes. - 4.3 Numerical Illustrations. - 4.4 Higher-Order Methods and Limiting. - 5 Non-uniform Grids - Finite Elements (1D). - 5.1 The Basic Galerkin Method. - 5.2 Standard Galerkin Error Estimates. - 5.3 Upwinding. - 6 Multi-dimensional Aspects. - 6.1 Cartesian Grid Discretizations. - 6.2 Diffusion on Cartesian Grids. - 6.3 Advection on Cartesian Grids. - 6.4 Transformed Cartesian Grids. - 6.5 Unstructured Grids. - 7 Notes on Moving Grids and Grid Refinement. - 7.1 Dynamic Regridding. - 7.2 Static Regridding. - IV SPLITTING METHODS. - 1 Operator Splitting. - 1.1 First-Order Splitting. - 1.2 Second-Order Symmetrical Splitting. - 1.3 Higher-Order Splittings. - 1.4 Abstract Initial Value Problems. - 1.5 Advection-Diffusion-Reaction Splittings. - 1.6 Dimension Splitting. - 1. 7 Boundary Values and Stiff Terms. - 2 LOD Methods. - 2.1 The LOD-Backward Euler Method. - 2.2 LOD Crank-Nicolson Methods. - 2.3 The Trapezoidal Splitting Method. - 2.4 Boundary Correction Techniques. - 2.5 Numerical Comparisons. - 3 ADI Methods. - 3.1 The Peaceman-Rachford Method. - 3.2 The Douglas Method. - 4 IMEX Methods. - 4.1 The IMEX-θ Method. - 4.2 IMEX Multistep Methods. - 4.3 Notes on IMEX Runge-Kutta Methods. - 4.4 Concluding Remarks and Tests. - 5 Rosenbrock AMF Methods. - 5.1 One-Stage Methods of Order One and Two. - 5.2 Two-Stage Methods of Order Two and Three. - 5.3 A Three-Stage Method of Order Two. - 5.4 Concluding Remarks and Tests. - 6 Numerical Examples. - 6.1 Two Chemo-taxis Problems from Biology. - 6.2 The Numerical Methods. - 6.3 Numerical Experiments. - V STABILIZED EXPLICIT RUNGE-KUTTA METHODS. - 1 The RKC Family. - 1.1 Stability Polynomials. - 1.2 Integration Formulas. - 1.3 Internal Stability and Full Convergence Properties. - 2 The ROCK Family. - 2.1 Stability Polynomials. - 2.2 Integration Formulas. - 2.3 Internal Stability and Convergence. - 3 Numerical Examples. - 3.1 A Combustion Model. - 3.2 A Radiation-Diffusion Model. - Bibliography. - Index.