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Optimal m ‐stage Runge‐Kutta schemes for steady‐state solutions of hyperbolic equations
Author(s) -
Chen MeiQin,
Chiu Chichia
Publication year - 1993
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690090604
Subject(s) - mathematics , discretization , eigenvalues and eigenvectors , nonlinear system , minimax , runge–kutta methods , convergence (economics) , hyperbolic partial differential equation , chebyshev filter , mathematical analysis , numerical analysis , mathematical optimization , partial differential equation , physics , quantum mechanics , economics , economic growth
A numerical scheme is developed to find optimal parameters and time step of m ‐stage Runge‐Kutta (RK) schemes for accelerating the convergence to ‐steady‐state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m ‐stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.