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In this paper, we construct first- and second-order fully discrete schemes for the Cahn–Hilliard–Hele–Shaw system based on the Fourier-spectral method for spatial discretization. For temporal discretization, we combine two efficient approaches, including the scalar auxiliary variable (SAV) method for linearizing nonlinear potentials and the zero-energy-contribution method (ZEC) for decoupling nonlinear couplings. These schemes are linear, fully decoupled, and unconditionally energy stable, requiring only the solution of a sequence of elliptic equations with constant coefficients at each time step. The rigorous proof of the error analysis for the first-order scheme is shown. In addition, several numerical examples are presented to demonstrate the stability, accuracy, and efficiency of the proposed scheme.

A novel numerical strategy is introduced for computing approximations of solutions to a Cahn-Hilliard model with degenerate mobilities. This model has recently been introduced as a second-order phase-field approximation for surface diffusion flows. Its numerical discretization renders challenging due to the degeneracy of the mobilities, which generally requires an implicit treatment to avoid stability issues at the price of increased complexity costs. To mitigate this drawback, we consider new first- and second-order Scalar Auxiliary Variable (SAV) schemes, which, differently from existing approaches, focus on the relaxation of the mobility, rather than the Cahn-Hilliard energy. These schemes are introduced and analysed theoretically in the general context of gradient flows and then specialised for the Cahn-Hilliard equation with mobilities. Various numerical experiments are conducted to highlight the advantages of these new schemes both in terms of accuracy, effectiveness and computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a second‐order accurate (in time) numerical scheme for the droplet liquid film coarsening model by using the scalar auxiliary variable (SAV) method, and the Fourier‐spectral method is used in space. The scheme is linear; thus, it could be solved very efficiently. Meanwhile, due to the application of SAV approach, the unconditional energy stability of the numerical scheme is also derived, without any restriction on the time step size. In addition, we also provide a rigorous error estimate that shows that our second‐order numerical scheme with Fourier‐spectral method in space converges with order OΔt2+hm$$ O\left(\Delta {t}^2+{h}^m\right) $$, where Δt$$ \Delta t $$ and h$$ h $$ are time and space step sizes, respectively. Finally, a few numerical experiments are performed, which confirms the efficiency and accuracy of the proposed scheme, including the tests of the convergence, the mass conservation and the decrease of the energy. The simulation of coarsening process with time is also observed. [ABSTRACT FROM AUTHOR]

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equations at each time step. By using the build-in unconditional stability of the GSAV approach, we derive optimal global (resp. local) in time error estimates in the two (resp. three) dimensional case for the velocity and pressure approximations. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete scheme is proposed to solve the nonlinear Schrödinger-Possion equations. The scheme is developed by the scalar auxiliary variable (SAV) approach, the Crank-Nicolson temproal discretization and the Galerkin-Legendre spectral spatial discretization. The fully discrete scheme is proved to be mass- and energy- conserved. Moreover, unconditional energy stability and convergence of the scheme are obtained by the use of the Gagliardo-Nirenberg and some Sobolev inequalities. Numerical results are presented to confirm our theoretical findings.

In this paper, we construct efficient schemes based on the scalar auxiliary variable block-centered finite difference method for the modified phase field crystal equation, which is a sixth-order nonlinear damped wave equation. The schemes are linear, conserve mass and unconditionally dissipate a pseudo energy. We prove rigorously second-order error estimates in both time and space for the phase field variable in discrete norms. We also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete scheme is proposed to solve the nonlinear Schrödinger-Possion equations. The scheme is developed by the scalar auxiliary variable (SAV) approach, the Crank-Nicolson temproal discretization and the Galerkin-Legendre spectral spatial discretization. The fully discrete scheme is proved to be mass- and energy- conserved. Moreover, unconditional energy stability and convergence of the scheme are obtained by the use of the Gagliardo-Nirenberg and some Sobolev inequalities. Numerical results are presented to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We construct new first- and second-order pressure correctionschemes using the scalar auxiliary variable approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates in the two dimensional case for the velocity and pressure approximation of the first-order scheme without any condition on the time step. [ABSTRACT FROM AUTHOR]

In this paper, we consider a second-order scalar auxiliary variable Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

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 that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried out to show that both velocity and pressure approximations are second-order accurate in time and space. Numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a linear, second order, energy and entropy-production-rate preserving scheme for a thermodynamically consistent phase field model for dentritic crystal growth, combining an energy quadratization strategy with the finite element method. The scheme can be decomposed into a series of Poisson equations for efficient numerical implementations. Numerical tests are carried out to verify the accuracy of the scheme and simulations are conducted to demonstrate the effectiveness of the scheme on benchmark examples. [ABSTRACT FROM AUTHOR]

We present in this paper construction and analysis of a blockcentered 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 the 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. [ABSTRACT FROM AUTHOR]

The scalar auxiliary variable (SAV) approach [31] and its generalized version GSAV proposed in [20] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in [21] for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV approach, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the k -th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for k = 1 , 2 , 3 , 4 , 5. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach. We construct a relaxed version of the generalized SAV (R-GSAV) approach with the following advantages: • it is unconditionally energy stable with respect to a modified energy that is directly linked to the original energy; • it can achieve high-order accuracy and can be applied to general dissipative systems; • it is numerically shown to provide improved accuracy compared with the GSAV approach. [ABSTRACT FROM AUTHOR]

• This paper addresses one unresolved issue for the SAV method. • It introduces a relaxation step for calculating the auxiliary variables. • The relaxation strategy improves the accuracy and consistency of the SAV method. The scalar auxiliary variable (SAV) method was introduced by Shen et al. in [36] and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulting from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach. [ABSTRACT FROM AUTHOR]

We propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz–Petrich model. The function space, initial and boundary conditions are carefully chosen so that it fixes the relative orientation and displacement, and we follow a gradient flow to let the interface find its optimal structure. The gradient flow is discretized by the scalar auxiliary variable (SAV) approach in time, and a spectral method in space using quasiperiodic Fourier series and generalized Jacobi polynomials. We use the method to study interface between striped, hexagonal and dodecagonal phases, especially when the interface is quasiperiodic. The numerical examples show that our method is efficient, accurate, and can successfully capture the interfacial structure. [ABSTRACT FROM AUTHOR]