Backward error analysis for variational discretisations of partial differential equations
R.I. McLachlan, C. Offen, ArXiv:2006.14172 (n.d.).
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Abstract
In backward error analysis, an approximate solution to an equation is
compared to the exact solution to a nearby "modified" equation. In numerical
ordinary differential equations, the two agree up to any power of the step
size. If the differential equation has a geometric property then the modified
equation may share it. In this way, known properties of differential equations
can be applied to the approximation. But for partial differential equations,
the known modified equations are of higher order, limiting applicability of the
theory. Therefore, we study symmetric solutions of discretized partial
differential equations that arise from a discrete variational principle. These
symmetric solutions obey infinite-dimensional functional equations. We show
that these equations admit second-order modified equations which are
Hamiltonian and also possess first-order Lagrangians in modified coordinates.
The modified equation and its associated structures are computed explicitly for
the case of rotating travelling waves in the nonlinear wave equation.
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arXiv:2006.14172
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Cite this
McLachlan RI, Offen C. Backward error analysis for variational discretisations of partial differential equations. arXiv:200614172.
McLachlan, R. I., & Offen, C. (n.d.). Backward error analysis for variational discretisations of partial differential equations. ArXiv:2006.14172.
@article{McLachlan_Offen, title={Backward error analysis for variational discretisations of partial differential equations}, journal={arXiv:2006.14172}, author={McLachlan, Robert I and Offen, Christian} }
McLachlan, Robert I, and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial Differential Equations.” ArXiv:2006.14172, n.d.
R. I. McLachlan and C. Offen, “Backward error analysis for variational discretisations of partial differential equations,” arXiv:2006.14172. .
McLachlan, Robert I., and Christian Offen. “Backward Error Analysis for Variational Discretisations of Partial Differential Equations.” ArXiv:2006.14172.