The Experts below are selected from a list of 14979 Experts worldwide ranked by ideXlab platform
Jie Shen - One of the best experts on this subject based on the ideXlab platform.
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Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations
Journal of Computational Physics, 2021Co-Authors: Xavier Antoine, Jie Shen, Qinglin TangAbstract:In this paper, based on the Scalar Auxiliary Variable (SAV) approach and a newly proposed Lagrange multiplier (LagM) approach originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.
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scalar Auxiliary Variable lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear schrodinger gross pitaevskii equations
2020Co-Authors: Xavier Antoine, Jie Shen, Qinglin TangAbstract:In this paper, based on the Scalar Auxiliary Variable (SAV) approach [40, 41] and a newly proposed Lagrange multiplier (LagM) approach [21, 20] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrodinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.
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a highly efficient and accurate new scalar Auxiliary Variable approach for gradient flows
SIAM Journal on Scientific Computing, 2020Co-Authors: Fukeng Huang, Jie Shen, Zhiguo YangAbstract: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...
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multiple scalar Auxiliary Variable msav approach and its application to the phase field vesicle membrane model
SIAM Journal on Scientific Computing, 2018Co-Authors: Qing Cheng, Jie ShenAbstract:We consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar Auxiliary Variable (SAV) approach, and we develop the multiple sc...
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convergence and error analysis for the scalar Auxiliary Variable sav schemes to gradient flows
SIAM Journal on Numerical Analysis, 2018Co-Authors: Jie ShenAbstract:We carry out convergence and error analysis of the scalar Auxiliary Variable (SAV) methods for $L^2$ and $H^{-1}$ gradient flows with a typical form of free energy. We first derive $H^2$ bounds, under certain assumptions suitable for both the gradient flows and the SAV schemes, which allow us to establish the convergence of the SAV schemes under mild conditions. We then derive error estimates with further regularity assumptions. We also discuss several other gradient flows, which cannot be cast in the general framework used in this paper, but for which convergence and error analysis can still be established using a similar procedure.
Zhiguo Yang - One of the best experts on this subject based on the ideXlab platform.
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a highly efficient and accurate new scalar Auxiliary Variable approach for gradient flows
SIAM Journal on Scientific Computing, 2020Co-Authors: Fukeng Huang, Jie Shen, Zhiguo YangAbstract: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...
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A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized Auxiliary Variable with guaranteed positivity
Journal of Computational Physics, 2020Co-Authors: Zhiguo Yang, Suchuan DongAbstract:Abstract We present a framework for devising discretely energy-stable schemes for general dissipative systems based on a generalized Auxiliary Variable. The Auxiliary Variable, a scalar number, can be defined in terms of the energy functional by a general class of functions, not limited to the square root function adopted in previous approaches. The current method has another remarkable property: the computed values for the generalized Auxiliary Variable are guaranteed to be positive on the discrete level, regardless of the time step sizes or the external forces. This property of guaranteed positivity is not available in previous approaches. A unified procedure for treating the dissipative governing equations and the generalized Auxiliary Variable on the discrete level has been presented. The discrete energy stability of the proposed numerical scheme and the positivity of the computed Auxiliary Variable have been proved for general dissipative systems. The current method, termed gPAV (generalized Positive Auxiliary Variable), requires only the solution of linear algebraic equations within a time step. With appropriate choice of the operator in the algorithm, the resultant linear algebraic systems upon discretization involve only constant and time-independent coefficient matrices, which only need to be computed once and can be pre-computed. Several specific dissipative systems are studied in relative detail using the gPAV framework. Ample numerical experiments are presented to demonstrate the performance of the method, and the robustness of the scheme at large time step sizes.
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numerical approximation of incompressible navier stokes equations based on an Auxiliary energy Variable
Journal of Computational Physics, 2019Co-Authors: Zhiguo Yang, Lianlei Lin, Suchuan DongAbstract:Abstract We present a numerical scheme for approximating the incompressible Navier-Stokes equations based on an Auxiliary Variable associated with the total system energy. By introducing a dynamic equation for the Auxiliary Variable and reformulating the Navier-Stokes equations into an equivalent system, the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation. Within each time step, the algorithm involves the computations of two pressure fields and two velocity fields by solving several de-coupled individual linear algebraic systems with constant coefficient matrices, together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost. A number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm.
Mejdi Azaiez - One of the best experts on this subject based on the ideXlab platform.
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a variant of scalar Auxiliary Variable approaches for gradient flows
Journal of Computational Physics, 2019Co-Authors: Dianming Hou, Mejdi AzaiezAbstract:Abstract In this paper, we propose and analyze a new class of schemes based on a variant of the scalar Auxiliary Variable (SAV) approaches for gradient flows. Precisely, we construct more robust first and second order unconditionally stable schemes by introducing a new defined Auxiliary Variable to deal with nonlinear terms in gradient flows. The new approach consists in splitting the gradient flow into decoupled linear systems with constant coefficients, which can be solved using existing fast solvers for the Poisson equation. This approach can be regarded as an extension of the SAV method; see, e.g., Shen et al. (2018) [21] , in the sense that the new approach comes to be the conventional SAV method when α = 0 and removes the boundedness assumption on ∫ Ω F ( ϕ ) d x required by the SAV. The new approach only requires that the total free energy or a part of it is bounded from below, which is more realistic in physically meaningful models. The unconditional stability is established, showing that the efficiency of the new approach is less restricted to particular forms of the nonlinear terms. A series of numerical experiments is carried out to verify the theoretical claims and illustrate the efficiency of our method.
Jiang Yang - One of the best experts on this subject based on the ideXlab platform.
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the scalar Auxiliary Variable sav approach for gradient flows
Journal of Computational Physics, 2018Co-Authors: Jie Shen, Jiang YangAbstract:Abstract We propose a new approach, which we term as scalar Auxiliary Variable (SAV) approach, to construct efficient and accurate time discretization schemes for a large class of gradient flows. The SAV approach is built upon the recently introduced IEQ approach. It enjoys all advantages of the IEQ approach but overcomes most of its shortcomings. In particular, the SAV approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. The scheme is not restricted to specific forms of the nonlinear part of the free energy, so it applies to a large class of gradient flows. Numerical results are presented to show that the accuracy and effectiveness of the SAV approach over the existing methods.
Qinglin Tang - One of the best experts on this subject based on the ideXlab platform.
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Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations
Journal of Computational Physics, 2021Co-Authors: Xavier Antoine, Jie Shen, Qinglin TangAbstract:In this paper, based on the Scalar Auxiliary Variable (SAV) approach and a newly proposed Lagrange multiplier (LagM) approach originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.
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scalar Auxiliary Variable lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear schrodinger gross pitaevskii equations
2020Co-Authors: Xavier Antoine, Jie Shen, Qinglin TangAbstract:In this paper, based on the Scalar Auxiliary Variable (SAV) approach [40, 41] and a newly proposed Lagrange multiplier (LagM) approach [21, 20] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrodinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.
