High-Order Multiderivative IMEX Schemes. (arXiv:2008.03370v2 [physics.comp-ph] UPDATED)
<a href="http://arxiv.org/find/physics/1/au:+Dittmann_A/0/1/0/all/0/1">Alexander J. Dittmann</a> (University of Maryland)
Recently, a 4th-order asymptotic preserving multiderivative implicit-explicit
(IMEX) scheme was developed (Sch”utz and Seal 2020, arXiv:2001.08268). This
scheme is based on a 4th-order Hermite interpolation in time, and uses an
approach based on operator splitting that converges to the underlying
quadrature if iterated sufficiently. Hermite schemes have been used in
astrophysics for decades, particularly for N-body calculations, but not in a
form suitable for solving stiff equations. In this work, we extend the scheme
presented in Sch”utz and Seal 2020 to higher orders. Such high-order schemes
offer advantages when one aims to find high-precision solutions to systems of
differential equations containing stiff terms, which occur throughout the
physical sciences. We begin by deriving Hermite schemes of arbitrary order and
discussing the stability of these formulas. Afterwards, we demonstrate how the
method of Sch”utz and Seal 2020 generalises in a straightforward manner to any
of these schemes, and prove convergence properties of the resulting IMEX
schemes. We then present results for methods ranging from 6th to 12th order and
explore a selection of test problems, including both linear and nonlinear
ordinary differential equations and Burgers’ equation. To our knowledge this is
also the first time that Hermite time-stepping methods have been applied to
partial differential equations. We then discuss some benefits of these schemes,
such as their potential for parallelism and low memory usage, as well as
limitations and potential drawbacks.
Recently, a 4th-order asymptotic preserving multiderivative implicit-explicit
(IMEX) scheme was developed (Sch”utz and Seal 2020, arXiv:2001.08268). This
scheme is based on a 4th-order Hermite interpolation in time, and uses an
approach based on operator splitting that converges to the underlying
quadrature if iterated sufficiently. Hermite schemes have been used in
astrophysics for decades, particularly for N-body calculations, but not in a
form suitable for solving stiff equations. In this work, we extend the scheme
presented in Sch”utz and Seal 2020 to higher orders. Such high-order schemes
offer advantages when one aims to find high-precision solutions to systems of
differential equations containing stiff terms, which occur throughout the
physical sciences. We begin by deriving Hermite schemes of arbitrary order and
discussing the stability of these formulas. Afterwards, we demonstrate how the
method of Sch”utz and Seal 2020 generalises in a straightforward manner to any
of these schemes, and prove convergence properties of the resulting IMEX
schemes. We then present results for methods ranging from 6th to 12th order and
explore a selection of test problems, including both linear and nonlinear
ordinary differential equations and Burgers’ equation. To our knowledge this is
also the first time that Hermite time-stepping methods have been applied to
partial differential equations. We then discuss some benefits of these schemes,
such as their potential for parallelism and low memory usage, as well as
limitations and potential drawbacks.
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