Method of lines transpose: high order L-stable \(\mathcal O(N)\) schemes for parabolic equations using successive convolution. (English) Zbl 1339.65199
Summary: 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 one-dimensional heat equation solver that uses fast \(\mathcal O(N)\) 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 multidimensional 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.
MSC:
| 65M80 | Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 35K55 | Nonlinear parabolic equations |
| 35K57 | Reaction-diffusion equations |
| 31A10 | Integral representations, integral operators, integral equations methods in two dimensions |
Keywords:
method of lines transpose; transverse method of lines; Rothe’s method; parabolic PDEs; implicit methods; boundary integral methods; alternating direction implicit methods; ADI schemes; higher order L-stable; multiderivativeReferences:
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