The existence and uniqueness of solutions for the nonlinear algebraic equations in implicit r-k methods

Abstract

In the past a few years, the problem of the existence and uniqueness of solutions to the nonlinear algebraic equations in implicit Runge-Kutta methods (IR-KMs) has been studied by many authors. In this paper, on a wide class of initial value problems (IVPs) we present a condition on the R-K matrix, which insures the unique solvability of the equations. It turns out that our result improves and generalizes previous ones. Some examples (including the Lobotto IIIC method) show that the problem of their unique solvability can be solved only by our method.

Keywords:

• IR-KMs
• nonlinear algebraic equations
• the existence and uniqueness of solutions to equations

AMS(M0S) subject classification:

• 65L05

C.R Categories:

• G.1.7

^§ Present Address: Math. Dept., The Univ. of Texas at Dallas Richardson, TX 75083-0688 U.S.A

^§ Present Address: Math. Dept., The Univ. of Texas at Dallas Richardson, TX 75083-0688 U.S.A

Notes

^§ Present Address: Math. Dept., The Univ. of Texas at Dallas Richardson, TX 75083-0688 U.S.A

Additional information

Notes on contributors

Zhang Jianguo

‡