Discrete Techniques for Numerical Relativity

Discrete Techniques for Numerical Relativity - Novel...

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Novel techniques for Numerical Relativity Manuel Tiglio, LSU (KITP Gravitation Conf 5/14/03) 1 Discrete techniques for Numerical Relativity Manuel Tiglio Hearne Institute for Theoretical Physics Louisiana State University In collaboration with G. Calabrese, L. Lehner, D. Neilsen, J. Pullin, O. Reula, O. Sarbach ITP, May 03 A far from complete list of numerical questions Choosing the system to solve : coordinate conditions, free or constrained evolution, boundary conditions. Once the problem has been defined, one needs to solve it in a numerically stable way. Numerical stability = convergence : the property that errors go way with resolution. Several issues that need to be considered for numerical stability: l How does one discretize near boundaries? l If doing extrapolation near a corner or a curvilinear boundary, in which direction should one do so? Does higher order extrapolation help? l If using cubic boxes as computational domains, how does one impose boundary conditions at corners and edges? l How shall one use dissipation near boundaries? l Can one avoid artificial, numerical errors that grow fast in time?
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Novel techniques for Numerical Relativity Manuel Tiglio, LSU (KITP Gravitation Conf 5/14/03) 2 Overview, main ideas l Method of lines : from semidiscrete to fully discrete stability. l Numerical stability as a discrete version of well posedness . l Numerical stability through discrete energy estimates : Summation by parts Boundary conditions l Going beyond numerical stability: preventing errors from growing . l Application: black hole excision . l Dynamical control of discrete constraint violations . l Multi-patch evolution. The method of lines l Given a set of differential equations, l Semidiscrete problem: first discretize space but not time: l Prove numerical stability for the semidiscrete problem (system of ODE’s). u t x u B u D t x u A u i i i t ) , , ( ) , , ( r r + = k j i k j i i k j i i k j i i k j i u t x u B u D t x u A u dt d , , , , , , , , , , ) , , ( ) , , ( + = l If now you use appropriate time integrators, stability for the fully discrete problem follows. The details of what you did in the semidiscrete case do not matter, provided that the semidiscrete problem was stable. l Get spatial discretizations that give semidiscrete stability, and time integrators that are “locally stable”. Combine them at will.
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Novel techniques for Numerical Relativity Manuel Tiglio, LSU (KITP Gravitation Conf 5/14/03) 3 Numerical stability: a discrete version of well posedness Two reasons for looking at numerical stability for linear problems: It is a necessary condition for stability in the non-linear case.
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Discrete Techniques for Numerical Relativity - Novel...

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