Affiliation:
1. Department of Computational Mathematics and Artificial Intelligence, RUDN University, 117198 Moscow, Russia
2. Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract
We consider the problem of determining the weights of difference schemes whose form is specified by a particular symbolic expression. The order of approximation of the differential equation is equal to a given number. To solve it, it was propose to proceed from considering systems of differential equations of a general form to one scalar equation. This method provides us with some values for the weights, which we propose to test using Richardson’s method. The method was shown to work in the case of low-order schemes. However, when transitioning from the scalar problem to the vector and nonlinear problems, the reduction of the order of the scheme, whose weights are selected for the scalar problem, occurs in different families of schemes. This was first discovered when studying the Shanks scheme, which belongs to the family of explicit Runge–Kutta schemes. This does not deteriorate the proposed strategy itself concerning the simplification of the weight-determination problem, which should include a clause on mandatory testing of the order using the Richardson method.
Funder
RUDN University Strategic Academic Leadership Program
Reference20 articles.
1. Stone, P. (2024, July 20). Maple Worksheets on the Derivation of Runge-Kutta Schemes. Available online: http://www.peterstone.name/Maplepgs/RKcoeff.html.
2. Hairer, E., Wanner, G., and Nørsett, S.P. (2008). Solving Ordinary Differential Equations I, Springer. [3rd ed.].
3. Runge-Kutta methods: Some historical notes;Butcher;Appl. Numer. Math.,1996
4. High-order explicit Runge-Kutta methods using m-symmetry;Feagin;Neural Parallel Sci. Comput.,2012
5. On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems;Baddour;Program. Comput. Soft.,2023