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

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

/pdf/the-immersed-boundary-method-xfkx3wd291.pdf

The immersed boundary method

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids (Berger & Aftosmis 1998, Clarke et al. 1986, Zeeuw & Powell 1991). These methods have been extended to simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus have capabilities similar to those of IB methods. In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli & Zaleski (1999). Consider the simulation of flow past a solid body shown in Figure 1a. The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries b is generated. This is then used as a boundary condition to generate a grid in the volume f occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines (Ferziger & Peric 1996). Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the

/pdf/immersed-boundary-methods-4id27kda4o.pdf

Immersed boundary methods

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

https://crd.lbl.gov/assets/pubs_presos/AMCS/ANAG/A113.pdf

Local adaptive mesh refinement for shock hydrodynamics

We present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Our approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, data structures and grid generation procedure, and conclude with numerical examples in one and two space dimensions.

Adaptive mesh refinement for hyperbolic partial differential equations

An Adaptive Version of the Immersed Boundary Method

This work documents a new method for rapid and robust Cartesian mesh generation for component-based geometry. The new algorithm adopts a novel strategy that first intersects the components to extract the wetted surface before proceeding with volume mesh generation in a second phase. The intersection scheme is based on a robust geometry engine that uses adaptive precision arithmetic and automatically and consistently handles geometric degenerades with an algorithmic tie-breaking routine. The intersection procedure has worst-case computational complexity of O(N log N) and is demonstrated on test cases with up to 121 overlapping and intersecting components, including a variety of geometric degeneracies. The volume mesh generation takes the intersected surface triangulation as input and generates the mesh through cell division of an initially uniform coarse grid. In refining hexagonal cells to resolve the geometry, the new approach preserves the ability to directionally divide cells that are well aligned with local geometry. The mesh generation scheme has linear asymptotic complexity with memory requirements that total approximately 14-17 words/cell. The mesh generation speed is approximately 10 6 cells/minute on a typical engineering workstation

Robust and efficient Cartesian mesh generation for component-based geometry

A local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described. The method is based on the use of local grid patches superimpo...

Marsha Berger

Papers

Local adaptive mesh refinement for shock hydrodynamics

Adaptive mesh refinement for hyperbolic partial differential equations

An Adaptive Version of the Immersed Boundary Method

Robust and efficient Cartesian mesh generation for component-based geometry

Three-dimensional adaptive mesh refinement for hyperbolic conservation laws