Analysis of the error of difference solutions as a basis for improving accuracy
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Vladimir V. Shaydurov
,
Lidiya V. Gileva
Abstract
First, we give a short history of four algorithmic approaches in computational mathematics based on the analysis and use of the asymptotic behavior of the error of an approximate solution where the mesh size of a difference grid tends to zero. These approaches are Richardson’s ‘extrapolation to the limit’, Runge’s accuracy rule, Romberg’s rule for calculating integrals, and improving the grid solutions by high-order differences. The first two approaches were initially developed based on the intuitive conclusion about the asymptotic behavior of the error back in the early 20th century. The last two algorithms, at the time of their appearance in the second half of the 20th century, already used theoretical results on the special asymptotic behavior of the error of quadrature rules or difference solutions.
The latter approach, despite its good computational efficiency for ordinary differential equations, has not yet been properly developed for solving multidimensional difference problems. Therefore, as an introductory illustration, we present a method for increasing the order of convergence in time for the solution of an initial boundary value problem for a parabolic equation by correcting the right-hand side with differences of a higher order. The increase in accuracy order is justified theoretically and demonstrated by a numerical example.
Funding statement: This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2025-1606).
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Preface
- Algorithms of variational data assimilation for problems of ocean dynamics
- Identifiability of basic models and parameters in mathematical immunology
- Analysis of the error of difference solutions as a basis for improving accuracy
- Numerical method for hydraulic fracture propagation in a two-phase poroelastoplasticity model
- The method of adjoint equations for solving the problem of quasi-geostrophic currents in a two-layer ocean periodic channel
Artikel in diesem Heft
- Frontmatter
- Preface
- Algorithms of variational data assimilation for problems of ocean dynamics
- Identifiability of basic models and parameters in mathematical immunology
- Analysis of the error of difference solutions as a basis for improving accuracy
- Numerical method for hydraulic fracture propagation in a two-phase poroelastoplasticity model
- The method of adjoint equations for solving the problem of quasi-geostrophic currents in a two-layer ocean periodic channel
