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Geometric numerical integrators with complex coefficients for solving differential equations
Prof. Sergio Blanes
Universidad Politecnica de Valencia
Abstract:
Geometric Numerical Integrators (GNIs) are numerical methods for solving differential equations which preserve most qualitative properties of the exact solution. Explicit GNIs have shown a high performance for solving many different problems. This is the case of separable Hamiltonian systems over very long integrations because they are easy to implement and usually show a high performance with low error propagations. However, these methods usually require some fractional backward time step, and they cannot be used on a number of important problems. We can circumvent this problem if complex coefficients are allowed to build GNIs. In general, to introduce complex coefficients makes the numerical schemes more costly, but there are many problems where this is not the case. In addition, one has to project to the real space, and this projection can destroy the geometric structure of the numerical solution as well as the error propagation for long time integrations. GNIs with complex coefficients can be seen as methods which provide the exact solution of perturbed problems evolving on a generalized higher dimensional manifold (backward error analysis). We present some preliminary results on the preservation of the geometric structure under projection, and this is illustrated on several numerical examples.
Tuesday, February 19, 2013
11:00AM AP&M 2402
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
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Prof. Sergio Blanes
Universidad Politecnica de Valencia
Abstract:
Geometric Numerical Integrators (GNIs) are numerical methods for solving differential equations which preserve most qualitative properties of the exact solution. Explicit GNIs have shown a high performance for solving many different problems. This is the case of separable Hamiltonian systems over very long integrations because they are easy to implement and usually show a high performance with low error propagations. However, these methods usually require some fractional backward time step, and they cannot be used on a number of important problems. We can circumvent this problem if complex coefficients are allowed to build GNIs. In general, to introduce complex coefficients makes the numerical schemes more costly, but there are many problems where this is not the case. In addition, one has to project to the real space, and this projection can destroy the geometric structure of the numerical solution as well as the error propagation for long time integrations. GNIs with complex coefficients can be seen as methods which provide the exact solution of perturbed problems evolving on a generalized higher dimensional manifold (backward error analysis). We present some preliminary results on the preservation of the geometric structure under projection, and this is illustrated on several numerical examples.
Tuesday, February 19, 2013
11:00AM AP&M 2402