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NUMERICAL COMPARISON OF MASS-CONSERVATIVE SCHEMES FOR THE GROSS-PITAEVSKII EQUATION

机译：总保守方案的数值比较总体保守方案

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摘要

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).
机译：在本文中，我们提出了各种量保守离散化的数值比较，用于时间依赖于时间依赖性的总质规则的毛诗歌。我们有三个主要目标。首先，我们希望澄清与另外节能或伴称的方法相比纯粹的大规模保守方法。其次，我们将比较彼此中的节能和辛方法的准确性。三，我们将研究线性化的节能方法是否与需要在每个时间步骤中解决全部非线性问题的方法的准确度而受到准确性的损失。为了获得代表性的比较，我们的数值实验涵盖了不同的物理相关的测试用例，例如行驶孤子，固定多粒子，Bose-Einstein在光学晶格和涡旋图案中凝结的凝结物，在快速旋转的超流体中。我们还应考虑一个涉及伪薄膜绝缘体的计算严重的测试用例。我们的空间离散化是基于整个纸张的有限元。我们还将特别注意长时间的行为和多阶段尺寸和网格尺寸之间的耦合条件。本文的主要观察是单独的大规模保护不会导致复杂环境中的竞争方法。此外，节能和辛的方法既可靠又准确，但节能方案在测试用例中达到明显更高的准确性。最后，在我们的实验中表现最佳的方案是一种节能弛豫方案，C. Besse（Sinum，42（3）：934-952,2004）提出了线性时间踩踏。

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