We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn and Cahn--Hilliard ...

Energy-Decaying Extrapolated RK--SAV Methods for the Allen--Cahn and Cahn--Hilliard Equations

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on...

The Exponential Scalar Auxiliary Variable (E-SAV) Approach for Phase Field Models and Its Explicit Computing

We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit l...

A Highly Efficient and Accurate New Scalar Auxiliary Variable Approach for Gradient Flows

/pdf/improving-the-accuracy-and-consistency-of-the-scalar-3gpai4m6.pdf

Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation

In this paper we propose and analyze a second order accurate (in time) numerical scheme for the square phase field crystal equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. Its primary difference with the standard phase field crystal model is an introduction of the 4-Laplacian term in the free energy potential, which in turn leads to a much higher degree of nonlinearity. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a second order backward differentiation formula is applied in the temporal stencil. Meanwhile, a direct application of the SAV method faces certain difficulties, due to the involvement of the 4-Laplacian term, combined with a derivation of the lower bound of the nonlinear energy functional. In the proposed numerical method, an appropriate decomposition for the physical energy functional is formulated, so that the nonlinear energy part has a well-established global lower bound, and the rest terms lead to constant-coefficient diffusion terms with positive eigenvalues. In turn, the numerical scheme could be very efficiently implemented by constant-coefficient Poisson-like type solvers (via FFT), and energy stability is established by introducing an auxiliary variable, and an optimal rate convergence analysis is provided for the proposed SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

Energy stability and convergence of SAV block-centered finite difference method for gradient flows

An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell (MAC) scheme is constructed for the Navier--Stokes equations. A particular feature of the scheme is th...

pdf/error-analysis-of-the-sav-mac-scheme-for-the-navier-stokes-440gut7mbo.pdf

Error Analysis of the SAV-MAC Scheme for the Navier--Stokes Equations

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt2 + N−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

/pdf/stability-and-error-estimates-of-the-sav-fourier-spectral-2607b79uj5.pdf

Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation

In this article, two characteristic block-centred finite difference schemes are introduced and analysed to solve the nonlinear convection-dominated diffusion equation. One scheme is a linear difference approximation which shows that the discrete and errors are , while the other is a two-grid scheme which demonstrates that the discrete and errors are , where h corresponds to a finer grid and H corresponds to a coarser grid. Error estimates with both schemes are established on a non-uniform rectangular grid. Finally, numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis and validate the efficiency of the two-grid method.

Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order $O(\Delta t^2+N^{-m})$, where $\Delta t$, $N$ and $m$ are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

Stability and Error estimates of the SAV Fourier-spectral method for the Phase Field Crystal Equation

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