2,301 research outputs found
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
A new approximate matrix factorization for implicit time integration in air pollution modeling
Implicit time stepping typically requires solution of one or several linear systems with a matrix I−τJ per time step where J is the Jacobian matrix. If solution of these systems is expensive, replacing I−τJ with its approximate matrix factorization (AMF) (I−τR)(I−τV), R+V=J, often leads to a good compromise between stability and accuracy of the time integration on the one hand and its efficiency on the other hand. For example, in air pollution modeling, AMF has been successfully used in the framework of Rosenbrock schemes. The standard AMF gives an approximation to I−τJ with the error τ2RV, which can be significant in norm. In this paper we propose a new AMF. In assumption that −V is an M-matrix, the error of the new AMF can be shown to have an upper bound τ||R||, while still being asymptotically . This new AMF, called AMF+, is equal in costs to standard AMF and, as both analysis and numerical experiments reveal, provides a better accuracy. We also report on our experience with another, cheaper AMF and with AMF-preconditioned GMRES
Unconditional Stability for Multistep ImEx Schemes: Theory
This paper presents a new class of high order linear ImEx multistep schemes
with large regions of unconditional stability. Unconditional stability is a
desirable property of a time stepping scheme, as it allows the choice of time
step solely based on accuracy considerations. Of particular interest are
problems for which both the implicit and explicit parts of the ImEx splitting
are stiff. Such splittings can arise, for example, in variable-coefficient
problems, or the incompressible Navier-Stokes equations. To characterize the
new ImEx schemes, an unconditional stability region is introduced, which plays
a role analogous to that of the stability region in conventional multistep
methods. Moreover, computable quantities (such as a numerical range) are
provided that guarantee an unconditionally stable scheme for a proposed
implicit-explicit matrix splitting. The new approach is illustrated with
several examples. Coefficients of the new schemes up to fifth order are
provided.Comment: 33 pages, 7 figure
High order operator splitting methods based on an integral deferred correction framework
Integral deferred correction (IDC) methods have been shown to be an efficient
way to achieve arbitrary high order accuracy and possess good stability
properties. In this paper, we construct high order operator splitting schemes
using the IDC procedure to solve initial value problems (IVPs). We present
analysis to show that the IDC methods can correct for both the splitting and
numerical errors, lifting the order of accuracy by with each correction,
where is the order of accuracy of the method used to solve the correction
equation. We further apply this framework to solve partial differential
equations (PDEs). Numerical examples in two dimensions of linear and nonlinear
initial-boundary value problems are presented to demonstrate the performance of
the proposed IDC approach.Comment: 33 pages, 22 figure
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