Polynomial Difference Schemes with Energy and Momentum Multipliers on Orthogonal Uniform Meshes: Stencil Restrictions and Monge--Ampère-Type Examples
Abstract
Experience with invariant finite-difference schemes for one-dimensional hydro\-dynamic-type equations for a function u(t,x) suggests that nonlinear polynomial schemes on orthogonal uniform meshes with the 9-point stencil cannot simultaneously preserve difference analogues of the energy and momentum conservation laws, with the multipliers given by the standard difference approximations of ut and ux. For the 9-point stencil this hypothesis is proved for polynomials of arbitrary degree. On larger stencils it is investigated by means of the difference analogue of the direct method: for polynomials of bounded degree, such schemes are found to exist only on sufficiently large stencils, and the second-order equations they approximate are of Monge--Ampère type. As a by-product, a scheme for the homogeneous Monge--Ampère equation is obtained that has five conservation laws, admits most of the symmetries of the differential equation, and is invariant under the discrete dihedral group D4. Numerical tests on exact solutions confirm its second order of approximation and show that the conservation laws are satisfied on these solutions with the same order of accuracy.
Related
- Discrete orthogonal polynomials related to Hahn difference operator
- Equidistribution, multipliers, and moving targets in polynomial families
- Geometric difference families and related variable-weight geometric orthogonal codes
- Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences
- Greedy Orthogonal Polynomial Manifolds with Lipschitz Control for Dimensionality Reduction and Reduced Modeling
BibTeX
@misc{kaptsov2026polynomial,
title = {{Polynomial Difference Schemes with Energy and Momentum Multipliers on Orthogonal Uniform Meshes: Stencil Restrictions and Monge--Ampère-Type Examples}},
author = {E. I. Kaptsov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08979},
eprint = {2610.08979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08979},
}
RIS
TY - UNPB
TI - Polynomial Difference Schemes with Energy and Momentum Multipliers on Orthogonal Uniform Meshes: Stencil Restrictions and Monge--Ampère-Type Examples
AU - E. I. Kaptsov
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08979
UR - https://arxiv.org/abs/2610.08979
AB - Experience with invariant finite-difference schemes for one-dimensional hydro\-dynamic-type equations for a function u(t,x) suggests that nonlinear polynomial schemes on orthogonal uniform meshes with the 9-point stencil cannot simultaneously preserve difference analogues of the energy and momentum conservation laws, with the multipliers given by the standard difference approximations of u_t and u_x. For the 9-point stencil this hypothesis is proved for polynomials of arbitrary degree. On larger stencils it is investigated by means of the difference analogue of the direct method: for polynomials of bounded degree, such schemes are found to exist only on sufficiently large stencils, and the second-order equations they approximate are of Monge--Ampère type. As a by-product, a scheme for the homogeneous Monge--Ampère equation is obtained that has five conservation laws, admits most of the symmetries of the differential equation, and is invariant under the discrete dihedral group D_4. Numerical tests on exact solutions confirm its second order of approximation and show that the conservation laws are satisfied on these solutions with the same order of accuracy.
ER -