Exact Finite-Difference Schemes

Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov, Sergey Lemeshevsky, Piotr Matus, Dmitriy Poliakov

• Éditeur: De Gruyter
• Année: 2016
• ISBN:
  9783110489729
• Édition: 1
• Sujet: Mathématiques
• Sous-sujet: Mathématiques appliquées

Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations.

Contents:
Basic notation
Preliminary results
Hyperbolic equations
Parabolic equations
Use of exact difference schemes to construct NSFD discretizations of differential equations
Exact and truncated difference schemes for boundary-value problem
Exact difference schemes for stochastic differential equations
Numerical blow-up time
Bibliography

Sergey Lemeshevsky, Piotr Matus, and Dmitriy Poliakov

Hyperbolic equations

Hyperbolic equations and systems include various linear and non-linear problems of mathematical physics. For many of them, it is possible to construct EDS and FDS of high order of approximation. In this chapter, we start with linear and semi-linear transport equations. Although they may seem to be rather simple, we are going to show general concepts of our technique (special Steklov averaging, non-standard weighing by spatial variables) to obtain EDSs. Let us note that constructed algorithms for semilinear equation are used for exact numerical simulation of shock waves with a variable amplitude. Obtained results are generalized to linear acoustics system, quasi-linear transport equations, second-order hyperbolic equations with boundary conditions of Dirichlet, Neumann or Robin types, and gas dynamics system.

1Linear and semi-linear transport equation

In this section, the FDSs for linear and semi-linear transport equations are described and analyzed.

Transport PDEs model the reactive transport of solutes in subsurface flows, fluid dynamics, and many other important applications. These equations admit solutions with moving steep fronts, which need to be resolved accurately in applications and often cause severe numerical difficulties. Standard finite-difference or finite-element methods of second order of accuracy construct num...