An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

Zhang, Min

doi:10.4208/cicp.oa-2018-0317

Cited by 20 publications

(29 citation statements)

References 42 publications

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“…where |||•||| denotes the maximum absolute value of the entries of a matrix and "∩" stands for matrix intersection. The reader is referred to [41] for the definition and geometric interpretation of matrix intersection. We take δ = 0.1 in our numerical simulations.…”

Section: Mmpde Moving Mesh Methodsmentioning

confidence: 99%

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A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

Zhang,

Huang,

Qiu

2020

Preprint

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A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the simulation of perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance and positivity-preserving properties, strategies are discussed in the use of slope limiting, positivity-preservation limiting, and data transferring between meshes. Particularly, it is suggested that a DG-interpolation scheme be used for the interpolation of both the flow variables and bottom topography from the old mesh to the new one and after each application of the positivity-preservation limiting on the water depth, a high-order correction be made to the approximation of the bottom topography according to the modifications in the water depth. Moreover, mesh adaptation based on the equilibrium variable and water depth is shown to give more desirable results than that based on the commonly used entropy function. Numerical examples in one and two spatial dimensions are presented to demonstrate the well-balance and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state.

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Section: Mmpde Moving Mesh Methodsmentioning

confidence: 99%

“…where φ i is the linear basis function associated with the vertex x i . We use a quasi-Lagrangian MM-DG method [27,41] to solve (3.1). The semi-discrete scheme for (3.1) is to find…”

Section: A Conservative Dg-interpolation Schemementioning

confidence: 99%

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

Zhang,

Huang,

Qiu

2020

Preprint

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“…At each data assimilation step, the common mesh is calculated by taking the metric tensor intersection of the monitor functions for each ensemble member; see, e.g., [28]. The metric tensor intersection is better understood in geometry as the intersection of unit balls in different norms that are associated with symmetric and positive definite (SPD) matrices.…”

Section: Forming the Common Meshmentioning

confidence: 99%

A Metric Tensor Approach to Data Assimilation with Adaptive Moving Meshes

Krause,

Huang,

Mechem

et al. 2021

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Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differential equations. However, special consideration must be given when combining adaptive meshing procedures with ensemblebased data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolves independently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines a framework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs) or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatial localization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh techniques can control and inform the placement of mesh points to concentrate near the location of observations, reducing the error of observation interpolation. This is especially beneficial when we have observations in locations that would otherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin (DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localization scheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observations are sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstrate the advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

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“…Luo et al considered a quasi-Lagrange moving mesh DG method (MMDG) for conservation laws [20] and multi-component flows [21]. Zhang et al studied the MMDG solution for the radiative transfer equation [26,27] and shallow water equations [28,29].…”

Section: Introductionmentioning

confidence: 99%

“…The objective of this work is to study these issues. To be specific, we consider a quasi-Lagrange MMDG method [20,26] with the Lax-Friedrichs (LF) flux for (1.1). A CFL condition for the L 1 stability of the MMDG method with P 0 elements is then established and analyzed for scalar linear equations (cf.…”

Section: Introductionmentioning

confidence: 99%

A study on CFL conditions for the DG solution of conservation laws on adaptive moving meshes

Zhang

Huang

Qiu

2021

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The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step has been based on CFL conditions established for fixed and uniform meshes. This work provides a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the α-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the α-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented.

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An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

Cited by 20 publications

References 42 publications

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

A Metric Tensor Approach to Data Assimilation with Adaptive Moving Meshes

A study on CFL conditions for the DG solution of conservation laws on adaptive moving meshes

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