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

Abstract

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.

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

“…Motivated by work done for parabolic equations (see e.g., [2,4]), we define the differential operator…”

“…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].…”

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

“…Over the past several years, the MOL T methods have been developed for solving the heat equation [4,6,20,24], Maxwell's equations [8], the advection equation and Vlasov equation [9], among others. This methodology can be generalized to solving some nonlinear problems, such as the Cahn-Hilliard equation [4]. However, it rarely applied to general nonlinear problems, mainly because efficient fast algorithms of inverting nonlinear BVPs are lacking and hence the advantage of the MOL T is compromised.…”

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

“…Moreover, a fast convolution algorithm is developed to ensure the computational complexity of the scheme is O(N ) [7,18,1], where N is the number of discrete mesh points. Over the past several years, the MOL T methods have been developed for solving the heat equation [4,6,24,20], Maxwell's equations [8], the advection equation and Vlasov equation [9], among others. This methodology can be generalized to solving some nonlinear problems, such as the Cahn-Hilliard equation [4].…”

“…Over the past several years, the MOL T methods have been developed for solving the heat equation [4,6,24,20], Maxwell's equations [8], the advection equation and Vlasov equation [9], among others. This methodology can be generalized to solving some nonlinear problems, such as the Cahn-Hilliard equation [4]. However, it rarely applied to general nonlinear problems, mainly because efficient fast algorithms of inverting nonlinear BVPs are lacking and hence the advantage of the MOL T is compromised.…”