A Program for the Numerical Solution of Large Sparse Systems of Algebraic and Implicitly Defined Stiff Differential Equations

1976 ◽  
Author(s):  
Richard Franke
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Stiff Differential Equations ◽  
Sparse Systems ◽  
Large Sparse Systems

The numerical solution of linear stiff differential equations

1972 ◽  
Vol 71 (3) ◽  
pp. 505-515 ◽  
Author(s):  
J. R. Cash
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Initial Conditions ◽  
Stiff Differential Equations ◽  
General Method

AbstractA general method is given and illustrated by application to particular cases for obtaining subdominant solutions of stiff difference and differential equations, i.e. when rapidly varying solutions – transients or otherwise – are possible but are in fact excluded by the initial conditions.

scholarly journals A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 914
Author(s):  
Zarina Bibi Ibrahim ◽  
Amiratul Ashikin Nasarudin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability And Convergence ◽  
Stiff Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

ACCURATE PARALLEL INTEGRATION OF LARGE SPARSE SYSTEMS OF DIFFERENTIAL EQUATIONS

1996 ◽  
Vol 06 (04) ◽  
pp. 535-568 ◽  
Author(s):  
DONALD J. ESTEP ◽  
ROY D. WILLIAMS
Keyword(s):  
Differential Equations ◽  
Error Estimation ◽  
Lorenz System ◽  
Sparse Matrix ◽  
Posteriori Error ◽  
Two Dimensions ◽  
Partial Differential ◽  
Sparse Systems ◽  
The Lorenz System ◽  
Large Sparse Systems

We describe a MIMD parallel code to solve a general class of ordinary differential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial differential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and flexibility of the software. We discuss the sources of error in solving differential equations, and the resulting constraints on time steps. We also discuss the theory of a posteriori error analysis for the Galerkin finite element methods, and its implementation in error control and estimation. The software is designed in a matrix-free fashion, which enables the solver to effectively tackle large sparse systems with minimal memory consumption and an easy and natural transition to MIMD (distributed memory) parallelism. In addition, there is no need for the choice of a particular representation of a sparse matrix. All memory is dynamically allocated, with a new expandable array object used for archiving results. The implicit solution of the discrete equations is carried out by replaceable modules: the nonlinear solver module may be a full Newton scheme or a quasi-Newton; these in turn are implemented with a linear solver, for which we have used both a direct solver and QMR, an iterative (Krylov space) method. Three computations are presented: the Lorenz system, which has dimension three and the discretized versions of the (partial-differential) bistable equation in one and two dimensions. The Lorenz system demonstrates the quality of the error estimation. The discretized bistable examples provide large sparse systems, and our precise error estimation shows, contrary to standard error estimates, that reliable computation is possible for large times.

Sign in / Sign up

Export Citation Format

Share Document