Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations
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E. I. Kaptsov
Abstract
When choosing suitable finite-difference schemes for hydrodynamic-type equations, preferences are given to various properties of schemes, such as their monotonicity, stability, conservation of phase volumes, etc. In the present report, we focus on the criterion of invariance of schemes, i. e., we consider finite-difference equations and meshes that preserve the symmetries of the original differential equations. However, this is not enough for the existence of conservation laws, both in the differential and finite-difference cases. In the case where the equations have a Lagrangian, the situation reduces to finding invariant Lagrangians. In Lagrangian coordinates, equations of hydrodynamic type are variational, and the construction of invariant difference schemes is often significantly simplified. In this case, uniform orthogonal meshes can be used, which preserve their geometric structure under the action of group transformations inherited from the original equations. On uniform orthogonal meshes, in many cases, it is possible to construct invariant conservative schemes that possess difference analogues of all local conservation laws of the original models. The report is devoted primarily to the practical aspects of constructing schemes of the described type. For this, a number of special techniques and methods have been developed. Various equations of the theory of shallow water and one-dimensional equations of magnetohydrodynamics are considered as examples.
Abstract
When choosing suitable finite-difference schemes for hydrodynamic-type equations, preferences are given to various properties of schemes, such as their monotonicity, stability, conservation of phase volumes, etc. In the present report, we focus on the criterion of invariance of schemes, i. e., we consider finite-difference equations and meshes that preserve the symmetries of the original differential equations. However, this is not enough for the existence of conservation laws, both in the differential and finite-difference cases. In the case where the equations have a Lagrangian, the situation reduces to finding invariant Lagrangians. In Lagrangian coordinates, equations of hydrodynamic type are variational, and the construction of invariant difference schemes is often significantly simplified. In this case, uniform orthogonal meshes can be used, which preserve their geometric structure under the action of group transformations inherited from the original equations. On uniform orthogonal meshes, in many cases, it is possible to construct invariant conservative schemes that possess difference analogues of all local conservation laws of the original models. The report is devoted primarily to the practical aspects of constructing schemes of the described type. For this, a number of special techniques and methods have been developed. Various equations of the theory of shallow water and one-dimensional equations of magnetohydrodynamics are considered as examples.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- About the editors IX
- List of contributors XI
- Intermediate systems and invariants 1
- Folding in fluids 19
- Vortex models of plane turbulent flows 37
- Equations of dyon electromagnetic field 53
- Minimal action principle for gravity and electrodynamics, Einstein lambda, and Lagrange points 65
- Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations 83
- On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium 111
- Approximate solution of a boundary-value problem for a model of the far momentumless turbulent wake 125
- Analysis of overdetermined system that describes the special class of two-dimensional motion of an ideal fluid 135
- Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters 143
- Index 165
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- About the editors IX
- List of contributors XI
- Intermediate systems and invariants 1
- Folding in fluids 19
- Vortex models of plane turbulent flows 37
- Equations of dyon electromagnetic field 53
- Minimal action principle for gravity and electrodynamics, Einstein lambda, and Lagrange points 65
- Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations 83
- On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium 111
- Approximate solution of a boundary-value problem for a model of the far momentumless turbulent wake 125
- Analysis of overdetermined system that describes the special class of two-dimensional motion of an ideal fluid 135
- Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters 143
- Index 165
