Integration of Partitioned Stiff Systems of Ordinary Differential Equations
نویسنده
چکیده
Abstract. Partitioned systems of ordinary differential equations are in qualitative terms characterized as monotonically max-norm stable if each sub-system is stable and if the couplings from one sub-system to the others are weak. Each sub-system of the partitioned system may be discretized independently by the backward Euler formula using solution values from the other sub-systems corresponding to the previous time step. The monotone max-norm stability guarantees this discretization to be stable. This so-called decoupled implicit Euler method is ideally suited for parallel computers. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations. This paper considers strategies and techniques for partitioning a system into a monotonically max-norm stable system. It also presents error bounds to be used in controlling stepsize, relaxation between sub-systems and the validity of the partitioning. Finally a realistic example is presented.
منابع مشابه
On second derivative 3-stage Hermite--Birkhoff--Obrechkoff methods for stiff ODEs: A-stable up to order 10 with variable stepsize
Variable-step (VS) second derivative $k$-step $3$-stage Hermite--Birkhoff--Obrechkoff (HBO) methods of order $p=(k+3)$, denoted by HBO$(p)$ are constructed as a combination of linear $k$-step methods of order $(p-2)$ and a second derivative two-step diagonally implicit $3$-stage Hermite--Birkhoff method of order 5 (DIHB5) for solving stiff ordinary differential equations. The main reason for co...
متن کاملSymplectic and symmetric methods for the numerical solution of some mathematical models of celestial objects
In the last years, the theory of numerical methods for system of non-stiff and stiff ordinary differential equations has reached a certain maturity. So, there are many excellent codes which are based on Runge–Kutta methods, linear multistep methods, Obreshkov methods, hybrid methods or general linear methods. Although these methods have good accuracy and desirable stability properties such as A...
متن کاملExplicit stabilized integration of stiff determinisitic or stochastic problems
Explicit stabilized methods for stiff ordinary differential equations have a long history. Proposed in the early 1960s and developed during 40 years for the integration of stiff ordinary differential equations, these methods have recently been extended to implicit-explicit or partitioned type methods for advection-diffusion-reaction problems, and to efficient explicit solvers for stiff mean-squ...
متن کاملMultirate Numerical Integration for Ordinary Differential Equations
Subject headings: Multirate time stepping / Local time stepping / Ordinary differential equations / Stiff differential equations / Asymptotic stability / High-order Rosenbrock methods / Partitioned Runge-Kutta methods / Mono-tonicity / TVD / Stability / Convergence. Het onderzoek dat tot dit proefschrift heeft geleid werd mede mogelijk gemaakt door een Peter Paul Peterichbeurs –verstrekt door d...
متن کاملMethods for Parallel Integration of Stiff Systems of ODEs
This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagonal sub-system. With one or several sub-systems allo...
متن کاملA Class of Linearly Implicit Numerical Methods for Solving Stiff Ordinary Differential Equations
We introduce ABC-schemes, a new class of linearly implicit one-step methods for numerical integration of stiff ordinary differential equation systems. Formulas of ABC-schemes invoke the Jacobian of differential system similary to the methods of Rosenbrock type, but unlike the latter they include also the square of the Jacobian matrix.
متن کامل