Adaptation based on interpolation errors for high order mesh refinement methods applied to conservation laws

Baeza, Antonio; Martı́nez-Gavara, Anna; Mulet, Pep

doi:10.1016/j.apnum.2011.09.008

Cited by 14 publications

(13 citation statements)

References 38 publications

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“…Our first dynamic test case for the proposed framework is given by the hyperbolic inviscid Burgers equation given by the following initial and boundary conditions 71 :…”

Section: Nonlinear Scalar Conservation Laws: Inviscid Burgers Equationmentioning

confidence: 99%

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

Maulik

San

Behera

2018

Numerical Methods in Fluids

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Summary We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.

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“…Our first dynamic test case for the proposed framework is given by the hyperbolic inviscid Burgers equation given by the following initial and boundary conditions 71 :…”

Section: Nonlinear Scalar Conservation Laws: Inviscid Burgers Equationmentioning

confidence: 99%

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

Maulik

San

Behera

2018

Numerical Methods in Fluids

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“…Within the Eulerian framework on Cartesian grids, such a problem can be conveniently circumvented by resorting to Adaptive-Mesh-Refinement (AMR) with local time stepping, see e.g. [8,7,4,3,14,45,39,91]. There, the mesh is forced to refine only when and where this is needed, while it is recoarsened as soon as the chosen refinement criterion is no longer satisfied.…”

Section: Two-dimensional Explosion Problemsmentioning

confidence: 99%

“…An alternative to overcome the global CFL condition consists in the development of numerical schemes that allow for time-accurate local time stepping (LTS), where each element has to obey only a less restrictive local CFL stability condition, hence using its own optimal local timestep. Therefore, many efforts have been devoted to the construction of high order accurate Eulerian schemes with time-accurate LTS, developing either discontinuous Galerkin finite element methods [48,42,82,65,53,62,47] or high order accurate finite volume schemes with LTS [8,7,89,16,4,3,14,47,45,39]. The finite volume schemes with LTS adopt mainly classical adaptive mesh refinement (AMR) techniques in space and time or block-clustered local time stepping algorithms.…”

Section: Introductionmentioning

confidence: 99%

High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes

Boscheri

Dumbser

Zanotti

2015

Journal of Computational Physics

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We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). It extends our previous investigations on high order Lagrangian finite volume schemes with LTS carried out in [36] in one space dimension. The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm.We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as ∼ 4.7.

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“…Moreover, since the coupling iteration is more rigorous than the TMR approach, this may be inefficient in terms of CPU time and memory requirement for solving largescale problems. In the last few years, high order adaptive mesh refinement has been wildly used for solving hyperbolic equations [16][17][18][19][20]. It is worth mentioning the fifth order adaptive finite difference WENO schemes by Shen and Qiu [17] and the high order adaptive finite volume ADER-WENO scheme by Dumbser et al [19,20].…”

Section: Introductionmentioning

confidence: 99%

An adaptive local grid refinement method for 2D diffusion equation with variable coefficients based on block-centered finite differences

Zhai

Weng

Feng

2015

Applied Mathematics and Computation

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Adaptation based on interpolation errors for high order mesh refinement methods applied to conservation laws

Cited by 14 publications

References 38 publications

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes

An adaptive local grid refinement method for 2D diffusion equation with variable coefficients based on block-centered finite differences

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