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Register a new userArbitrarily High-order Unconditionally Energy Stable Schemes for Gradient Flow Models Using the Scalar Auxiliary Variable Approach
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Added by
Jia Zhao
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a novel family of high-order numerical schemes for the gradient flow models based on the scalar auxiliary variable (SAV) approach, which is named the high-order scalar auxiliary variable (HSAV) method. The newly proposed schemes could be shown to reach arbitrarily high order in time while preserving the energy dissipation law without any restriction on the time step size (i.e., unconditionally energy stable). The HSAV strategy is rather general that it does not depend on the specific expression of the effective free energy, such that it applies to a class of thermodynamically consistent gradient flow models arriving at semi-discrete high-order energy-stable schemes. We then employ the Fourier pseudospectral method for spatial discretization. The fully discrete schemes are also shown to be unconditionally energy stable. Furthermore, we present several numerical experiments on several widely-used gradient flow models, to demonstrate the accuracy, efficiency and unconditionally energy stability of the HSAV schemes. The numerical results verify that the HSAV schemes can reach the expected order of accuracy, and it allows a much larger time step size to reach the same accuracy than the standard SAV schemes.
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We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization (EQ) method, We formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models.
We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to $t_n=n Delta t$ with $Delta t$ the time step size, we linearize the EQ-reformulated gradient flow model in $(t_n, t_{n+1}]$ by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in $(t_n, t_{n+1}]$. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on prediction-correction methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.
We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance.
We present unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved without imposing any restriction on time steps and meshes. We achieve this in two steps. First, taking advantage of the penalty free DG method introduced by Liu and Yin [J Sci. Comput. 77:467--501, 2018] for spatial discretization, we reformulate an extended linearized ODE system by the energy quadratization (EQ) approach. Second, we apply an s-stage algebraically stable RK method for temporal discretization. The resulting fully discrete DG schemes are linear and unconditionally energy stable. In addition, we introduce a prediction-correction procedure to improve both the accuracy and stability of the scheme. We illustrate the effectiveness of the proposed schemes by numerical tests with benchmark problems.
In this paper, we develop a new class of high-order energy-preserving schemes for the Korteweg-de Vries equation based on the quadratic auxiliary variable technique, which can conserve the original energy of the system. By introducing a quadratic auxiliary variable, the original system is reformulated into an equivalent form with a modified quadratic energy, where the way of the introduced variable naturally produces a quadratic invariant of the new system. A class of Runge-Kutta methods satisfying the symplectic condition is applied to discretize the reformulated model in time, arriving at arbitrarily high-order schemes, which naturally conserve the modified quadratic energy and the produced quadratic invariant. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In order to match the high order precision of time, the Fourier pseudo-spectral method is employed for spatial discretization, resulting in fully discrete energy-preserving schemes. To solve the proposed nonlinear schemes effectively, we present a very efficient practically-structure-preserving iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed schemes. This new class of numerical strategies is rather general so that they can be readily generalized for any conservative systems with a polynomial energy.
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