发表机构
Suranaree University of Technology（苏拉纳里理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明9点模板多项式差分格式无法同时保持能量和动量守恒，并在更大模板上构造了具有五个守恒律的Monge–Ampère型格式，数值验证其二阶精度。
AI 中文摘要
对一维流体动力学型方程（函数$u(t,x)$）的不变有限差分格式的经验表明，在具有9点模板的正交均匀网格上，非线性多项式格式无法同时保持能量和动量守恒律的差分模拟，其中乘子由$u_t$和$u_x$的标准差分近似给出。对于9点模板，这一假设对任意次数的多项式均得到证明。在更大的模板上，通过直接方法的差分模拟进行研究：对于有界次数的多项式，此类格式仅存在于足够大的模板上，且它们逼近的二阶方程属于Monge–Ampère类型。作为副产品，得到了一个齐次Monge–Ampère方程的格式，该格式具有五个守恒律，容许微分方程的大部分对称性，并在离散二面体群$D_4$下不变。对精确解的数值测试确认了其二阶逼近精度，并表明在这些解上守恒律以相同精度阶得到满足。
英文摘要
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.
Comments46 pages, 3 figures, 4 tables, 2 appendices
