An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

Tao, Zhanjing; Jiang, Yan; Cheng, Yingda

doi:10.1016/j.jcp.2020.109770

Cited by 10 publications

(19 citation statements)

References 22 publications

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“…For nonlinear convection terms in the KdV equation (1.1) and the ZK equation (1.2), we use the interpolatory multiwavelets based on Hermite interpolations introduced in [24]. The treatment of the nonlinear convection terms is the same as that in the adaptive multiresolution DG scheme for solving conservation laws in [17].…”

Section: Using the Notation Ofmentioning

confidence: 99%

“…The treatment of the nonlinear convection terms is the same as that in the adaptive multiresolution DG scheme for solving conservation laws in [17]. For saving space, we omit the details and refer readers to [24,17].…”

Section: Using the Notation Ofmentioning

confidence: 99%

“…Two classes of multiwavelets are employed in our schemes to attain MRA. First, the Alperts orthonormal multiwavelets are adopted as the DG basis functions [25], and then the interpolatory multiwavelets [24] are introduced to deal with the nonlinear terms in the scheme.…”

Section: Introductionmentioning

confidence: 99%

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A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations

Huang¹,

Liu²,

Liu³

et al. 2021

Preprint

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In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Kortewegde Vries (KdV) equation and its two dimensional generalization, the Zakharov-Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed for KdV equation in [7]. For the ZK equation which contains mixed derivative terms, we develop a new UWDG formulation. The L 2 stability and the optimal error estimate with a novel local projection are established for this new scheme on regular meshes. Adaptivity is achieved based on multiresolution and is particularly effective for capturing solitary wave structures. Various numerical examples are presented to demonstrate the accuracy and capability of our methods.

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Section: Using the Notation Ofmentioning

confidence: 99%

Section: Using the Notation Ofmentioning

confidence: 99%

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A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations

Huang¹,

Liu²,

Liu³

et al. 2021

Preprint

Self Cite

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“…To represent a nonlinear function, it is natural to consider interpolation or collocation methods [31,2]. This is achieved by using sparse grid collocation methods introduced in [49], which gives a framework to design adaptive sparse grid collocation onto arbitrary high order piecewise polynomial spaces. We approximate the nonlinear terms in the semi-discrete DG scheme for (1) by a linear combination of collocation basis functions up to required order of accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of this paper is organized as follows. In Section 2, we review MRA associated with two sets of basis functions, i.e., the Alpert's multiwavelets [3] and the interpolatory multiwavelets [49]. The adaptive multiresolution DG scheme is constructed in Section 3 using both sets of multiwavelets.…”

Section: Introductionmentioning

confidence: 99%

An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions

Huang

Cheng

2019

Preprint

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In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations [25,26], a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration of accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantages of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests.

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Adaptive Sparse Grid Discontinuous Galerkin Method: Review and Software Implementation

Huang¹,

Guo²,

Cheng³

2023

Commun. Appl. Math. Comput.

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An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

Cited by 10 publications

References 22 publications

A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations

A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations

An adaptive multiresolution discontinuous Galerkin method with artificial viscosity for scalar hyperbolic conservation laws in multidimensions

Adaptive Sparse Grid Discontinuous Galerkin Method: Review and Software Implementation

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