Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

Gao, Qinjiao; Zhang, Shenggang

doi:10.3390/a9040086

Cited by 10 publications

(8 citation statements)

References 33 publications

(69 reference statements)

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“…Subsequently, scholars did further research on this basis and produced a large number of optimization methods and solving methods for nonlinear equations [45,46]. Saheya et al constructed an improved RALND function and proposed a class of quasi-Newton method based on RALND function [47]. Wang applied RALND function to nonlinear least squares problem and proposed a class of improved Gauss-Newton method based on RALND function [46].…”

Section: Improvement In Other Aspectsmentioning

confidence: 99%

Remarks on Solving Methods of Nonlinear Equations

Huang,

Zhao

2024

J. Mod. Mech. Eng. Technol.

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In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions.

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Section: Improvement In Other Aspectsmentioning

confidence: 99%

Remarks on Solving Methods of Nonlinear Equations

Huang,

Zhao

2024

J. Mod. Mech. Eng. Technol.

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“…The moving mesh method [17,18] is one of the popular adaptive methods and has been successfully applied to various problems that contain time-dependent localized singularities [19][20][21]. It usually tries to find a time-dependent one-to-one coordinate transformation between the physical domain and the computational domain, by solving an additional system of moving mesh partial differential equation (MMPDE), which equidistributes a certain monitor function of the physical solution [22,23].…”

Section: Introductionmentioning

confidence: 99%

Heat Conduction Simulation of 2D Moving Heat Source Problems Using a Moving Mesh Method

Liu

2020

Advances in Mathematical Physics

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This paper focuses on efficiently numerical investigation of two-dimensional heat conduction problems of material subjected to multiple moving Gaussian point heat sources. All heat sources are imposed on the inside of material and assumed to move along some specified straight lines or curves with time-dependent velocities. A simple but efficient moving mesh method, which continuously adjusts the two-dimensional mesh dimension by dimension upon the one-dimensional moving mesh partial differential equation with an appropriate monitor function of the temperature field, has been developed. The physical model problem is then solved on this adaptive moving mesh. Numerical experiments are presented to exhibit the capability of the proposed moving mesh algorithm to efficiently and accurately simulate the moving heat source problems. The transient heat conduction phenomena due to various parameters of the moving heat sources, including the number of heat sources and the types of motion, are well simulated and investigated.

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“…A mesh equation is usually used to compute node speeds in order to move the mesh. This has the advantage that one is not required to transfer the solution between meshes because the PDE is reformulated to take into account the fact that the nodes are moving [49,42,41,43,26,15]. Furthermore, these methods are thought to do a good job of reducing "dispersive errors", a property that is useful for this problem.…”

Section: Introductionmentioning

confidence: 99%

An hr-adaptive method for the cubic nonlinear Schrödinger equation

Mackenzie

Mekwi

2020

Journal of Computational and Applied Mathematics

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The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods (r-adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Mesh refinement methods (h-adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method (hr-adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user's input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.

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Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

Cited by 10 publications

References 33 publications

Remarks on Solving Methods of Nonlinear Equations

Remarks on Solving Methods of Nonlinear Equations

Heat Conduction Simulation of 2D Moving Heat Source Problems Using a Moving Mesh Method

An hr-adaptive method for the cubic nonlinear Schrödinger equation

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