The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls

被引:24
作者
Ascher, UM [1 ]
Reich, S
机构
[1] Univ British Columbia, Inst Appl Math, Vancouver, BC V6T 1Z4, Canada
[2] Univ British Columbia, Dept Comp Sci, Vancouver, BC V6T 1Z4, Canada
[3] Konrad Zuse Zentrum, D-14195 Berlin, Germany
关键词
midpoint scheme; Hamiltonian systems; highly oscillatory problems; decoupling; stability; differential-algebraic equations; molecular dynamics; multibody systems;
D O I
10.1137/S1064827597316059
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The (implicit) midpoint scheme, like higher-order Gauss-collocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or structural mechanics, because there is no stability restriction when it is applied to a simple harmonic oscillator. Although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations, one might still hope for good results when resonance-type instabilities are avoided. In this paper we investigate the suitability of the midpoint scheme for highly oscillatory, frictionless mechanical systems, where the step size k is much larger than the system's small parameter epsilon, in the case that the solution remains bounded as epsilon --> 0. We show that in general one must require that k(2)/epsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably (especially when fast and slow modes are tightly coupled) or, worse, the computation may yield misleading information. In some cases this may already happen when k = O(epsilon). The same holds for higher-order collocation at Gaussian points. The encountered restrictions on k are typically still better than the corresponding ones for explicit schemes.
引用
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页码:1045 / 1065
页数:21
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