Volume 5, Issue 1
A Geometric Space-Time Multigrid Algorithm for the Heat Equation

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 110-130.

Published online: 2012-05

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• Abstract

We study the time-dependent heat equation on its space-time domain that is discretised by a $k$-spacetree. $k$-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint. The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation's elliptic operator with a multiscale solution propagation in time. While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded, the holistic approach promises to exhibit a better parallel scalability than classical time stepping, adaptive dynamic refinement in space and time fall naturally into place, as well as the treatment of periodic boundary conditions of steady cycle systems, on-time computational steering is eased as the algorithm delivers guesses for the solution's long-term behaviour immediately, and, finally, backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

• Keywords

Adaptive Cartesian grids, geometric multiscale methods, heat equation, octree, spacetree, space-time discretisation.

65M50, 65M55, 65N50, 65N55, 65M22

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@Article{NMTMA-5-110, author = {}, title = {A Geometric Space-Time Multigrid Algorithm for the Heat Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {1}, pages = {110--130}, abstract = {

We study the time-dependent heat equation on its space-time domain that is discretised by a $k$-spacetree. $k$-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint. The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation's elliptic operator with a multiscale solution propagation in time. While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded, the holistic approach promises to exhibit a better parallel scalability than classical time stepping, adaptive dynamic refinement in space and time fall naturally into place, as well as the treatment of periodic boundary conditions of steady cycle systems, on-time computational steering is eased as the algorithm delivers guesses for the solution's long-term behaviour immediately, and, finally, backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.m12si07}, url = {http://global-sci.org/intro/article_detail/nmtma/5931.html} }
TY - JOUR T1 - A Geometric Space-Time Multigrid Algorithm for the Heat Equation JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 110 EP - 130 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.m12si07 UR - https://global-sci.org/intro/article_detail/nmtma/5931.html KW - Adaptive Cartesian grids, geometric multiscale methods, heat equation, octree, spacetree, space-time discretisation. AB -

We study the time-dependent heat equation on its space-time domain that is discretised by a $k$-spacetree. $k$-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint. The paper presents a full approximation storage geometric multigrid implementation for this setting that combines the smoothing properties of multigrid for the equation's elliptic operator with a multiscale solution propagation in time. While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded, the holistic approach promises to exhibit a better parallel scalability than classical time stepping, adaptive dynamic refinement in space and time fall naturally into place, as well as the treatment of periodic boundary conditions of steady cycle systems, on-time computational steering is eased as the algorithm delivers guesses for the solution's long-term behaviour immediately, and, finally, backward problems arising from the adjoint equation benefit from the the solution being available for any point in space and time.

Tobias Weinzierl & Tobias Köppl. (2020). A Geometric Space-Time Multigrid Algorithm for the Heat Equation. Numerical Mathematics: Theory, Methods and Applications. 5 (1). 110-130. doi:10.4208/nmtma.m12si07
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