Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings

Numerical Initial Value Problems in Ordinary Differential Equations

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry

A Higher-Order Boundary Treatment for Cartesian-Grid Methods

Stability regions of explicit "linear" time discretization methods for solving initial value problems are treated. If an integration method needsm function evaluations per time step, then we scale the stability region by dividing bym. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for "general" nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods.

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Stability of explicit time discretizations for solving initial value problems

In this paper, an extension of the waveform relaxation algorithm for solving large systems of ordinary differential equations is presented. The waveform relaxation algorithm is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces. Allowing the subspaces to overlap, i.e., dropping the assumption of disjointedness, an extension of this algorithm is obtained. This new algorithm, the so-called multisplitting algorithm, is also well suited for parallel computation. As numerical examples demonstrate, this overlapping of the subsystems heavily reduces the computation time.

/pdf/waveform-relaxation-with-overlapping-splittings-4y9ea157uh.pdf

Waveform relaxation with overlapping splittings

The A-contractivity of Runge-Kutta methods with respect to an inner product norm was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is extended to contractivity in a region bounded by a circle through the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point. It is shown that an explicit Runge-Kutta method cannot be contractive in any circle of this class if it is more than fourth order accurate.

/pdf/generalized-disks-of-contractivity-for-explicit-and-implicit-4op2motmbc.pdf

Generalized disks of contractivity for explicit and implicit Runge-Kutta methods

This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general `method-free' statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.

Rolf Jeltsch

Papers

A Higher-Order Boundary Treatment for Cartesian-Grid Methods

Stability of explicit time discretizations for solving initial value problems

Waveform relaxation with overlapping splittings

Generalized disks of contractivity for explicit and implicit Runge-Kutta methods

Stability and accuracy of time discretizations for initial value problems