Circuit simulation programs have proven to be most important computer-aided design tools for the analysis of the electrical performance of integrated circuits. One of the most common analyses performed by circuit simulators and the most expensive in terms of computer time is nonlinear time-domain transient analysis. Conventional circuit simulators were designed initially for the cost-effective analysis of circuits containing a few hundred transistors or less. Because of the need to verify the performance of larger circuits, many users have successfully simulated circuits containing thousands of transistors despite the cost. Recently, a new class of algorithms has been applied to the electrical IC simulation problem. New simulators using these methods provide accurate waveform information with up to two orders of magnitude speed improvement for large circuits. These programs use relaxation methods for the solution of the set of ordinary differential equations, which describe the circuit under analysis, rather than the direct sparse-matrix methods on which standard circuit simulators are based. In this paper, the techniques used in relaxation-based electrical simulation are presented in a rigorous and unified framework, and the numerical properties of the various methods are explored. Both the advantages and the limitations of these techniques for the analysis of large IC's are described

Relaxation-Based Electrical Simulation

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.

Waveform relaxation: theory and practice

An Investigation of Stability in the Large Behaviour of a Control Surface with Structural Non-Linearities in Supersonic Flow

— In this paper we present discretization and decomposition methods for a multi-component transport model of a chemical vapor deposition (CVD) process. CVD processes are used to manufacture deposition layers or bulk materials. In our transport model we simulate the deposition of thin layers. The microscopic model is based on the heavy particles, which are derived by approximately solving a linearized multicomponent Boltzmann equation. For the drift-process of the particles we propose diffusion-reaction equations as well as for the effects of heat conduction. We concentrate on solving the diffusion-reaction equation with analytical and numerical methods. For the chemical processes, modelled with reaction equations, we propose decomposition methods and decouple the multi-component models to simpler systems of differential equations. In the numerical experiments we present the computational results of our proposed models. Keywords— Chemical reactions, chemical vapor deposition, convection-diffusion-reaction equations, decomposition methods, multi-component transport.

Simulation of a Multi-Component Transport Model for the Chemical Reaction of a CVD-Process

Piecewise-linear (PWL) functions are frequently used to describe the nonlinear branch equations of nonlinear devices in LSI circuits. New techniques for the solution of the differential equations describing the behavior of piecewise-linear circuits will be presented. These techniques are based on the waveform relaxation method to decouple the system equations and Laplace transform techniques to solve the decoupled equations. Several desirable features of the resulting algorithm are discussed.

R. John Kaye

Papers

Solution of piecewise-linear ordinary differential equations using waveform relaxation and laplace transforms