Abstract
We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines.
Highlights
Many physical phenomena are modeled by means of nonlinear time evolutionary partial differential equations
A well-known procedure to carry out the numerical approximations is the so-called method of lines which is based on separating the spatial approximation, carried out by finite elements or other classical methods, from the time integration which is made with schemes for ordinary differential schemes [1,2]
In this work we are mainly interested in the time integration taking into account two basic properties of the ordinary differential system obtained after the spatial discretization: the stiffness and the nonlinearity
Summary
Many physical phenomena are modeled by means of nonlinear time evolutionary partial differential equations. These mathematical models are usually very complex and the numerical analysis is essential in order to obtain quantitative and qualitative information of the solution. A good choice is given by the Rosenbrock methods which have been previously considered to time discretize partial differential equations, mainly when they are semilinear [3,4,5,6,7,8,9] In such a case, the implicitness of the step is given through linear systems and not nonlinear ones, as would happen with Runge-Kutta methods
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