Methods for constructing invariant conservative finite-difference schemes for hydrodynamic-type equations

Abstract: When choosing suitable finite-difference schemes for equations of hydrodynamic type, preference is 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.

For equations of the hydrodynamic type, the construction of invariant difference schemes is often significantly simplified if the equations are considered in Lagrange coordinates. In this case, uniform orthogonal meshes can be used, which retain their geometric structure under the action of group transformations inherited from the original equations. In addition, in Lagrangian coordinates, it is easier to find conservation laws both for differential equations and for the corresponding invariant difference schemes. In a number of 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 primarily devoted to the practical aspects of designing schemes of the described type. For this, a number of special techniques and methods have been developed. The most convenient is the finite-difference analogue of the direct method, as well as the technique of constructing schemes based on approximations of conservation laws. Various equations of the theory of shallow water and one-dimensional equations of magnetohydrodynamics are considered as examples.

References

1. Dorodnitsyn V. A., Kaptsov E. I., Discrete shallow water equations preserving symmetries and conservation laws. J. Math. Phys., 62(8):083508, 2021.

2. Kaptsov E. I., Dorodnitsyn V. A., Meleshko S. V., Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates. Stud. Appl. Math., 2022; 149: 729–761.

3. Dorodnitsyn V. A., Kaptsov E. I., and Meleshko S. V., Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green–Naghdi equations. J. Nonlinear Math. Phys., 28:90–107, 2020.

4. Cheviakov A. F., Dorodnitsyn V. A., Kaptsov E. I., Invariant conservation law-preserving discretizations of linear and nonlinear wave equations, J. Math. Phys., 61 (2020) P. 081504.

5. Dorodnitsyn V. A., Kaptsov E. I., Invariant finite-difference schemes for plane one-dimensional MHD flows that preserve conservation laws. Mathematics, 10(8):1250, 2022.

6. Kaptsov E. I., Dorodnitsyn V. A., Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates. Submitted. Preprint: https://arxiv.org/abs/2304.03488.

7. Kaptsov E. I., Dorodnitsyn V. A., Meleshko S. V., Invariant finite-difference schemes for cylindrical one-dimensional MHD flows with conservation laws preservation. Submitted. Preprint: http://dx.doi.org/10.48550/arXiv.2302.05280.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

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Mathematical models and integration methods

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