WAVEFORM RELAXATION: THEORY AND PRACTICE

This paper surveys the family of Waveform Relaxation Methods for solving large systems of ordinary nonlinear differential equations. The basic W R algorithm will be reviewed, and many of the derivative algorithms will be presented, along with new convergence proofs. In addition, examples will be' analyzed that illustrate several of the implementation techniques used to improve the efficiency of the basic W R algorithm, along with theoretical results that indicate the strengths or limitations of these techniques. INTRODUCTION The tremendous increase in complexity of engineering design and availability of computing resources has made computer simulation an important and heavily used tool for both research and engineering design. Since many simulation problems are formulated as large systems of nonlinear ordinary differential equations (ODE'S). much research work has been devoted to solving ODE systems efficiently The standard approach to solving ODE systems is based on three techniques 111. [21 : i) Stiffly stable implicit integration methods, such as the Backward difference formulas, which convert the differential equations describing the system into a sequence of nonlinear algebraic equations. ii) Modified Newton methods to solve the algebraic equations by solving a sequence of linear problems. iii) Sparse Guassian Elimination to solve the systems of linear equations generated by the Newton method.