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Invariant conservation law-preserving discretizations of linear and nonlinear wave equations. (English) Zbl 1454.65057

Summary: Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils using the theory of Lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws. For a class of nonlinear wave equations arising in hyperelasticity, a nine-point implicit scheme is constructed, preserving four-point symmetries and three local conservation laws. Other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed.
©2020 American Institute of Physics

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65J08 Numerical solutions to abstract evolution equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
17B81 Applications of Lie (super)algebras to physics, etc.
74B20 Nonlinear elasticity

Software:

GeM
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References:

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