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Investigation and application of point implicit Runge–Kutta methods to inviscid flow problems
Author(s) -
Langer Stefan
Publication year - 2011
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2561
Subject(s) - runge–kutta methods , inviscid flow , mathematics , convergence (economics) , preconditioner , ansatz , rate of convergence , residual , flow (mathematics) , function (biology) , mathematical analysis , numerical analysis , mathematical optimization , iterative method , computer science , algorithm , geometry , classical mechanics , physics , mathematical physics , channel (broadcasting) , evolutionary biology , economics , biology , economic growth , computer network
SUMMARY We derive and investigate point implicit Runge–Kutta methods to significantly improve the convergence rate to approximate steady‐state solutions of inviscid flows. It turns out that the point implicit Runge–Kutta can be interpreted as a preconditioned explicit Runge–Kutta method, where the preconditioner arises naturally as local derivative of the residual function. Moreover, many preconditioners suggested in the literature so far are identified as special case of our general ansatz. Conditions will be formulated such that explicit Runge–Kutta methods with local time stepping are equivalent to point implicit methods. In numerical examples, we will demonstrate the improved convergence rates. Copyright © 2011 John Wiley & Sons, Ltd.