Practical Construction of Modified Hamiltonians

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.