Backward error analysis for multi-symplectic integration methods

被引：133

作者：

Moore, B

Reich, S ^{[1
]}

机构：

[1] Univ London Imperial Coll Sci Technol & Med, Dept Math, London, England

[2] Univ Surrey, Dept Math & Stat, Guildford GU2 5XH, Surrey, England

来源：

NUMERISCHE MATHEMATIK | 2003年 / 95卷 / 04期

关键词：

D O I：

10.1007/s00211-003-0458-9

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations obtained through backward error analysis, and using symplectic integration for Hamiltonian ODEs provides more incite into the modified equations. In this paper, the ideas of symplectic integration are extended to Hamiltonian PDEs, and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize, but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. These results are also applied to the nonlinear wave equation.

引用

页码：625 / 652

页数：28

共 24 条

[1] ON THE HAMILTONIAN INTERPOLATION OF NEAR-TO-THE-IDENTITY SYMPLECTIC MAPPINGS WITH APPLICATION TO SYMPLECTIC INTEGRATION ALGORITHMS [J].

BENETTIN, G ;

GIORGILLI, A .

JOURNAL OF STATISTICAL PHYSICS, 1994, 74 (5-6) :1117-1143

[2] Multi-symplectic structures and wave propagation [J].

Bridges, TJ .

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1997, 121 :147-190

[3] Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity [J].

Bridges, TJ ;

Reich, S .

PHYSICS LETTERS A, 2001, 284 (4-5) :184-193

[4] Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry [J].

Bridges, TJ ;

Derks, G .

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1999, 455 (1987) :2427-2469

[5] A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities [J].

Bridges, TJ .

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1997, 453 (1962) :1365-1395

[6]

DEUFLHARD P, 1999, LECT NOTES COMPUTATI, V4, P2

[7] The numerical integration of relative equilibrium solutions.: The nonlinear Schrodinger equation [J].

Durán, A ;

Sanz-Serna, JM .

IMA JOURNAL OF NUMERICAL ANALYSIS, 2000, 20 (02) :235-261

[8]

Fornberg B., 1998, PRACTICAL GUIDE PSEU

[9] Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit [J].

Friesecke, G ;

Pego, RL .

NONLINEARITY, 1999, 12 (06) :1601-1627

[10] On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry [J].

Gonzalez, O ;

Simo, JC .

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 1996, 134 (3-4) :197-222

← 1 2 3 →