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.

中文翻译：

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线性和非线性波动方程的不变守恒定律离散化

利用差分方程的李点对称理论和守恒定律构造的离散直接乘子法，获得了五点和九点模板上线性和非线性一维波动方程的对称和守恒守恒有限差分离散。特别是，对于线性波动方程，提出了一个明确的五点方案，该方案保留了其基本几何点对称性的离散模拟和六个相应的守恒定律。针对超弹性中产生的一类非线性波动方程，构造了一个九点隐式格式，保留了四点对称性和三个局部守恒定律。讨论了保留不同守恒定律子集的非线性波动方程的其他离散化。