Multisymplectic integrator
In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrators are geometric integrators, meaning that they preserve the geometry of the problems; in particular, the numerical method preserves energy and momentum in some sense, similar to the partial differential equation itself. Examples of multisymplectic integrators include the Euler box scheme and the Preissman box scheme.
Multisymplectic equations[]
A partial differential equation (PDE) is said to be a multisymplectic equation if it can be written in the form
where is the unknown, and are (constant) skew-symmetric matrices and denotes the gradient of .[1] This is a natural generalization of , the form of a Hamiltonian ODE.[2]
Examples of multisymplectic PDEs include the nonlinear Klein–Gordon equation , or more generally the nonlinear wave equation ,[3] and the KdV equation .[4]
Define the 2-forms and by
where denotes the dot product. The differential equation preserves symplecticity in the sense that
Taking the dot product of the PDE with yields the local conservation law for energy:
The local conservation law for momentum is derived similarly:
The Euler box scheme[]
A multisymplectic integrator is a numerical method for solving multisymplectic PDEs whose numerical solution conserves a discrete form of symplecticity.[7] One example is the Euler box scheme, which is derived by applying the symplectic Euler method to each independent variable.[8]
The Euler box scheme uses a splitting of the skewsymmetric matrices and of the form:
For instance, one can take and to be the upper triangular part of and , respectively.[9]
Now introduce a uniform grid and let denote the approximation to where and are the grid spacing in the time- and space-direction. Then the Euler box scheme is
where the finite difference operators are defined by
The Euler box scheme is a first-order method,[8] which satisfies the discrete conservation law
Preissman box scheme[]
Another multisymplectic integrator is the Preissman box scheme, which was introduced by Preissman in the context of hyperbolic PDEs.[12] It is also known as the centred cell scheme.[13] The Preissman box scheme can be derived by applying the Implicit midpoint rule, which is a symplectic integrator, to each of the independent variables.[14] This leads to the scheme
where the finite difference operators and are defined as above and the values at the half-integers are defined by
The Preissman box scheme is a second-order multisymplectic integrator which satisfies the discrete conservation law
Notes[]
- ^ Bridges 1997, p. 1374; Leimkuhler & Reich 2004, p. 335–336.
- ^ Bridges & Reich 2001, p. 186.
- ^ Leimkuhler & Reich 2004, p. 335.
- ^ Leimkuhler & Reich 2004, p. 339–340.
- ^ Bridges & Reich 2001, p. 186; Leimkuhler & Reich 2004, p. 336.
- ^ a b Bridges & Reich 2001, p. 187; Leimkuhler & Reich 2004, p. 337–338.
- ^ Bridges & Reich 2001, p. 187; Leimkuhler & Reich 2004, p. 341.
- ^ a b Moore & Reich 2003.
- ^ Moore & Reich 2003; Leimkuhler & Reich 2004, p. 337.
- ^ Moore & Reich 2003; Leimkuhler & Reich 2004, p. 342.
- ^ Moore & Reich 2003; Leimkuhler & Reich 2004, p. 343.
- ^ Bridges & Reich (2001, p. 190) refers to Abbott & Basco (1989) for the work by Preissman.
- ^ Islas & Schober 2004, pp. 591–593.
- ^ a b Bridges & Reich 2001, p. 190; Leimkuhler & Reich 2004, p. 344.
- ^ Bridges & Reich 2001, Thm 1; Leimkuhler & Reich 2004, p. 345.
References[]
- Abbott, M.B.; Basco, D.R. (1989), Computational Fluid Dynamics, Longman Scientific.
- Bridges, Thomas J. (1997), "A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities" (PDF), Proc. R. Soc. Lond. A, 453 (1962): 1365–1395, doi:10.1098/rspa.1997.0075.
- Bridges, Thomas J.; Reich, Sebiastian (2001), "Multi-Symplectic Integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity", Phys. Lett. A, 284 (4–5): 184–193, CiteSeerX 10.1.1.46.2783, doi:10.1016/S0375-9601(01)00294-8.
- Leimkuhler, Benedict; Reich, Sebastian (2004), Simulating Hamiltonian Dynamics, Cambridge University Press, ISBN 978-0-521-77290-7.
- Islas, A.L.; Schober, C.M. (2004), "On the preservation of phase space structure under multisymplectic discretization", J. Comput. Phys., 197 (2): 585–609, doi:10.1016/j.jcp.2003.12.010.
- Moore, Brian; Reich, Sebastian (2003), "Backward error analysis for multi-symplectic integration methods", Numer. Math., 95 (4): 625–652, CiteSeerX 10.1.1.163.8683, doi:10.1007/s00211-003-0458-9.
- Numerical differential equations