Watanabe, Tsuguhiro HIDM, a general purpose algorithm for nonlinear problems described by partial differential equations. (English) Zbl 0853.65086 Kawarada, H. (ed.) et al., Proceedings of the international conference on nonlinear mathematical problems in industry, held at Iwaki Meisei University, Iwaki, Japan, Nov. 9-13, 1992. I. Tokyo: Gakkōtosho, GAKUTO Int. Ser., Math. Sci. Appl. 1, 189-204 (1993). Summary: A new numerical scheme HIDM (higher-order implicit difference method) is developed to solve comprehensively algebraic equations, ordinary differential equations, partial differential equations and coupled systems of these equations under high accuracy and high numerical stability. High accuracy of the scheme is realized by higher-order discretization method and high numerical stability of the scheme is obtained by implicit difference method and tournament multidivided shooting scheme.A general purpose computer code HIDM2D is constructed based on the numerical scheme for variety of the time evolution of boundary value problems (1D spatial variable + time). High performance of the code is demonstrated by solving nonlinear dissipative equations, nonlinear dispersive equations, and nonlinear dynamics of unstable systems.For the entire collection see [Zbl 0830.00023]. MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations Keywords:nonlinear evolution equations; higher-order implicit difference method; numerical stability; tournament multidivided shooting scheme; nonlinear dissipative equations; nonlinear dispersive equations; nonlinear dynamics; unstable systems Software:HIDM2D PDFBibTeX XMLCite \textit{T. Watanabe}, GAKUTO Int. Ser., Math. Sci. Appl. 1, 189--204 (1993; Zbl 0853.65086)