Abstract
For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test its numerical performance through some experiments. The main advantage with respect to other implicit methods is that our approach has a global convergence. The other approaches need to ensure convergence of the iterative scheme used to approximate the associated nonlinear equations that appear for the implicitness. Notice that these iterative methods, for these nonlinear equations, have bounded basins of attraction.
Highlights
We start with the following stiff differential problem: x (t) =f ( x (t)), x (0) = x0 .Due to its stiffness, it is convenient to use implicit Runge-Kutta methods for the time integration step
A difficulty appeared related to the approximation of the nonlinear Runge-Kutta equations by iterative methods [2], since we need to find good initial guesses
The variational approach that we will use is based on the error functional: E( x ) =
Summary
We start with the following stiff differential problem:. = x0. It is convenient to use implicit Runge-Kutta methods for the time integration step. The variational approach that we will use is based on the error functional: E( x ) =. The error functional in (1) is associated with the original problem in a natural way: it only has a local minimum that is the solution of the problem. This property will imply the global convergence of our approach based on optimality conditions and minimization schemes like (steepest) descent methods. We only need to approximate linear problems; we can use the well-developed theory of the convergence of Runge-Kutta methods for stiff linear problems. The main goal of this article is to test the variational approach exposed before to solve this kind of problems
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