Using Broyden Updates to Approximate the Jacobian in a Semi-Implicit Backward Differentiation Code
Abstract
If one wishes to solve a system of stiff ordinary differential equations (ODEs), which are generally large and sparse, one requires the solution of a nonlinear equation involving the ODE Jacobian at each time step. This Jacobian is often formed by finite differences. Here Broyden's Method is discussed which will update the Jacobian as we go using information available in order to avoid the cost of recomputing the Jacobian. The update is a rank-one update. This method is implemented in the code STRUT. STRUT is the original code STEP which implements an Adams predictor-corrector for solving nonstiff problems coupled with Stewart's implementation of a Semi-Implicit Backward Differentiation Formula (SIBDF) predictor-corrector for solving stiff problems. Two models are discussed for implementing Broyden's Method over different time steps. A Method of Lines semidiscretization of Burgers' equation and a reaction-diffusion model are used as our test problems. Results are given for each model and are compared with the original STRUT code, LSODE and LSODA. (jhd)
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1989
- Accession Number
- ADA213239
Entities
People
- Laura D. Knight