Practical construction of modified Hamiltonians
Practical construction of modified Hamiltonians
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DOI：
10.1137/s106482750138318x
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发表时间：
2001-12-18
影响因子：
3.1
通讯作者：
Hardy, DJ
中科院分区：
文献类型：
--
作者：
Skeel, RD;Hardy, DJ
分析微分方程组数值解中离散化误差影响的最有效方法之一是检查修正方程，修正方程是（近似）离散解精确满足的方程。这些实际上并不存在，而是由离散化参数的幂的渐近展开定义的。尽管如此，如果扩展被适当地截断，所得到的修改后的方程具有非常接近离散解的解。在常微分方程的哈密顿系统的情况下，修改的方程也是哈密顿的当且仅当积分器是辛的。通过计算修改后的哈密顿量和监测它们如何守恒，可以获得特定计算的哈密顿量存在的证据。此外，能量漂移所造成的数值不稳定性更好地揭示了评估修改后的哈密顿。进行这种计算通常是复杂的，并且高度依赖于方法的细节，即使差异被用来近似导数。这里提出了一个相对简单的程序，几乎独立的积分器的内部结构，获得高度准确的估计修改后的哈密顿量。作为构造方法的一个好处，在二次汉密尔顿的情况下，修改后的汉密尔顿通过数值解完全守恒。
One of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the modified equations, which are equations that are exactly satisfied by the ( approximate) discrete solution. These do not actually exist in general but rather are defined by an asymptotic expansion in powers of the discretization parameter. Nonetheless, if the expansion is suitably truncated, the resulting modified equations have a solution which is remarkably close to the discrete solution. In the case of a Hamiltonian system of ordinary differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. Evidence for the existence of a Hamiltonian for a particular calculation is obtained by calculating modified Hamiltonians and monitoring how well they are conserved. Also, energy drifts caused by numerical instability are better revealed by evaluating modified Hamiltonians. Doing this calculation would normally be complicated and highly dependent on the details of the method, even if differences are used to approximate derivatives. A relatively simple procedure is presented here, nearly independent of the internal structure of the integrator, for obtaining highly accurate estimates for modified Hamiltonians. As a bonus of the method of construction, the modified Hamiltonians are exactly conserved by a numerical solution in the case of a quadratic Hamiltonian.