The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs; methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them.
In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions; Practical stability analysis in the presence of the boundaries and interfaces; Treatment of problems with different temporal/spatial scales either explicit or implicit; preservation of symmetries and additional constraints; physical regularization of singularities; resolution enhancement using adaptive mesh refinement and moving meshes.
- Self contained presentation of key issues in successful numerical simulation- Accessible to scientists and engineers with diverse background- Provides analysis of the dispersion relation, symmetries, particular solutions and instabilities of the partial differential equations
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1. Overview of Properties Partial Differential equations 2. Methods of Discretization 3. Convergence Theory for Initial Value Problems 4. Numerical Boundary Conditions 5. Problems with Multiple Temporal and Spatial Scales 6. Nonuniform, Adaptive and Moving Grids