Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

✍ Scribed by K. Dekker

• Publisher: Elsevier Science
• Year: 1977
• Tongue: English
• Weight: 938 KB
• Volume: 3
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(77)80013-7

✦ Synopsis

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.

~2 -"

does not depend on Y2" Second order, m-point formulas using two or three arrays of storage are constructed. In the latter case the resulting stability condition reads m-1 h ng~, modd.

(1.3)

Here o(J) denotes the spectral radius of the Jacobian matrix J. The number of derivative evaluations per time step for these formulas, however, is less than m,