Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

Causley, Matthew F.; Cho, Hanna; Christlieb, Andrew; Seal, David C.

doi:10.1137/15m1035094

Cited by 22 publications

(34 citation statements)

References 45 publications

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“…Semi-discrete scheme for 1D CH equation. We utilize the MOL T by discretizing (6) in time as the Backward Euler (BE) scheme,…”

Section: 1mentioning

confidence: 99%

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Causley¹,

Cho²,

Christlieb³

2017

SIAM J. Sci. Comput.

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Abstract. In this work, we develop an O(N ) implicit real space method in 1D and 2D for the Cahn Hilliard (CH) and vector Cahn Hilliard (VCH) equations, based on the Method Of Lines Transpose (MOL T ) formulation. This formulation results in a semi-discrete time stepping algorithm, which we prove is gradient stable in the H −1 norm. The spatial discretization follows from dimensional splitting, and an O(N ) matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high order, logically Cartesian (line-by-line) update. Our method is fast, but not restricted to periodic boundaries like the fast Fourier transform (FFT).The basic solver is implemented using the Backward Euler formulation, and we extend this to both backward difference (BDF) stencils, implicit Runge Kutta (SDIRK) and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH, and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that metastable states and ripening events can be simulated both quickly and efficiently.

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“…Semi-discrete scheme for 1D CH equation. We utilize the MOL T by discretizing (6) in time as the Backward Euler (BE) scheme,…”

Section: 1mentioning

confidence: 99%

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Causley¹,

Cho²,

Christlieb³

2017

SIAM J. Sci. Comput.

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“…To approximate the integral equations in BVP, the fast multipole method(FMM) solved the heat, Navier-Stokes and linearized Poisson-Boltmann equation in [16,22], Fourier-continuation alternating-direction(FC-AD) algorithm yields unconditionally stability from O(N 2 ) to O(N log N ) [1,24] and Causley et al [5] reduces the computational complexity of the method from O(N 2 ) to O(N ). A variety of schemes, based on the MOL T formulation, have been developed for solving a range of time-dependent PDEs, including the wave equation [3], the heat equation (e.g., the Allen-Cahn equation [4] and Cahn-Hilliard equation [2]), Maxwell's equations [6], and the Vlasov equation [9].…”

Section: Introductionmentioning

confidence: 99%

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

Christlieb

Guo

Jiang

2019

Journal of Computational Physics

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In [11], the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.Recent work on the MOL T has involved extending the method to solve more general nonlinear PDEs, for which an integral solution is generally not applicable. This work includes the nonlinear degenerate convection-diffusion equations [10], as well as the H-J equations [11]. The key idea of these papers involved exploiting the linearity of a given differential operator, rather than requiring linearity in the underlying equations. This allowed derivative operators in the problems to be expressed through kernel representations developed for linear problems. Formulating applicable derivative operators in this way ultimately facilitated the stability of the schemes, since a global coupling was introduced through the integral operator. As part of this embedding process, a kernel parameter β was introduced, and through a careful selection, was shown to yield schemes which are A-stable. Remarkably, it was shown that one could couple these representations for the derivative operators with an explicit time-stepping method, such as the strong-stability-preserving Runge-Kutta (SSP-RK) methods [17] and still obtain schemes which maintain unconditional stability [10,11]. To address shock-capturing and control non-physical oscillations, the latter two papers introduced quadrature formulas based on WENO reconstruction, along with a nonlinear filter.This paper seeks to extend the work in [10,11] to the H-J equations (1.1) defined on non-uniformly distributed spatial domains. In particular, several improvements are given. First, we develop the MOL T for mapped grids using a general coordinate transformation function, which allows for a non-uniform distribution of grid points. We show that, with this mapping, our numerical scheme is able to preserve the conservation property for the derivative of the solution to the H-J equation. We also describe a novel WENO-based quadrature for the spatial discretization, which uses a basis consisting of exponential polynomials, to improve the shock capturing capabilities of the method. Another difference in this paper, compared to...

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“…The MOL has been successfully applied to solve various types of problems in engineering and science. ()…”

Section: Introductionmentioning

confidence: 99%

“…The MOL has been successfully applied to solve various types of problems in engineering and science. [31][32][33][34][35] In this paper, a suitable set of basis functions for spatial discretization is constructed by using Jacobi polynomials. Moreover, the right and left space-fractional derivatives of the basis functions are derived in closed form, which help us to simplify the spatial discretization of the problem.…”

Section: Introductionmentioning

confidence: 99%

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

Almoaeet

Shamsi

Khosravian-Arab

et al. 2019

Math Methods in App Sciences

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We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.

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Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

Cited by 22 publications

References 45 publications

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

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