Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

Zouraris, Georgios E.

doi:10.1051/m2an/2020077

Cited by 7 publications

(3 citation statements)

References 15 publications

(36 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We are not aware of any analytical results concerning the convergence of the fully discrete method for the nonlinear Schrödinger equation, but for the semilinear heat equation an analysis was recently provided in [38].…”

Section: Relaxation Methods (Re-fem)mentioning

confidence: 99%

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2019

Kinetic &Amp; Related Models

View full text Add to dashboard Cite

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM,42(3):934-952,2004).

show abstract

Section: Relaxation Methods (Re-fem)mentioning

confidence: 99%

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2019

Kinetic &Amp; Related Models

View full text Add to dashboard Cite

show abstract

“…One particularly important aspect is that the analytical equation (1.2) conserves the total energy of the system and it was numerically observed [34] that an analogous discrete energy conservation can be a crucial property of numerical schemes to get reliable approximations in practical situations. Time integrators that conserve a modified energy are for example the popular Besse relaxation scheme [15,16,52] and the family of exponential Runge-Kutta schemes proposed in [26]. Time integrators that conserve the exact energy (up to spatial discretization errors) are more rare and include the Crank-Nicolson method based on the averaging of densities [3,10,32,35,45] and the continuous Galerkin time stepping proposed in [39].…”

Section: Introductionmentioning

confidence: 99%

Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

Döding¹,

Henning²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Practically, the discrete conservation of mass and energy is subject to the choice of the time integrator. Among others, mass conservative time discretizations have been studied in [57,61], time integrators that are mass conservative and symplectic are investigated in [3,25,51,54,56], energy conservative time discretizations in [34] and time discretization that preserve mass and energy simultaneously are addressed in [3,7,9,11,12,16,31,33,50,62]. For further discretizations we refer to [5,8,10,36,53] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of time invariants for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2020

Preprint

View full text Add to dashboard Cite

This paper considers the numerical treatment of the time-dependent Gross-Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture the time invariants with an accuracy of order O(H 6 ) with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the L ∞ (L 2 )-norm with order O(τ 2 + H 4 ), where τ denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

show abstract

Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

Cited by 7 publications

References 15 publications

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

Superconvergence of time invariants for the Gross-Pitaevskii equation

Contact Info

Product

Resources

About