Stable and Accurate Filtering Procedures

Lundquist, Tomas; Nordström, Jan

doi:10.1007/s10915-019-01116-9

Cited by 13 publications

(22 citation statements)

References 10 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that Proposition 2 above can be seen as a straightforward extension of Proposition 1 in [19], which relates to explicit filtering procedures. In the context of filters however, we have P n+1 = P n , which makes the assumption that (8) holds superfluous.…”

Section: Proofmentioning

confidence: 92%

“…On the semi-discrete level, we consider a single AMR operation at time t = t n as a transmission problem [19,22]…”

Section: Amr As a Transmission Problemmentioning

confidence: 99%

“…In Sect. 2 we outline a general theory of AMR stability based on IPP operators, mirroring and expanding some of the key results for numerical filters in [19]. In Sect.…”

Section: Introductionmentioning

confidence: 98%

“…semi-discretized systems of equations coupled in time using interpolation. Focusing on the example of numerical filters in [19], it was demonstrated that the stability of transmission problems in general is tightly linked to the IPP concept discussed above. In this work, we will therefore consider the use of IPP type operators for time dependent AMR interpolation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stable Dynamical Adaptive Mesh Refinement

2021

Self Cite

View full text Add to dashboard Cite

We consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

show abstract

Section: Proofmentioning

confidence: 92%

“…On the semi-discrete level, we consider a single AMR operation at time t = t n as a transmission problem [19,22]…”

Section: Amr As a Transmission Problemmentioning

confidence: 99%

“…In Sect. 2 we outline a general theory of AMR stability based on IPP operators, mirroring and expanding some of the key results for numerical filters in [19]. In Sect.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stable Dynamical Adaptive Mesh Refinement

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, Lundquist and Nordström [11] removed the ad hoc nature of filtering in finite difference (FD) methods. They derive a contractivity condition on the explicit filter matrix by re-framing the filtering procedure as a transmission problem [13].…”

Section: Introductionmentioning

confidence: 99%

Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

Nordström

Winters

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

show abstract

A realizable filtered intrusive polynomial moment method

Alldredge¹,

Frank

Kusch

et al. 2022

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Intrusive uncertainty quantification methods for hyperbolic problems exhibit spurious oscillations at shocks, which leads to a significant reduction of the overall approximation quality. Furthermore, a challenging task is to preserve hyperbolicity of the gPC moment system. An intrusive method which guarantees hyperbolicity is the intrusive polynomial moment (IPM) method, which performs the gPC expansion on the entropy variables. The method, while still being subject to oscillations, requires solving a convex optimization problem in every spatial cell and every time step.The aim of this work is to mitigate oscillations in the IPM solution by applying filters. Filters reduce oscillations by damping high order gPC coefficients. Naive filtering, however, may lead to unrealizable moments, which means that the IPM optimization problem does not have a solution and the method breaks down. In this paper, we propose and analyze two separate strategies to guarantee the existence of a solution to the IPM problem. First, we propose a filter which maintains realizability by being constructed from an underlying Fokker-Planck equation. Second, we regularize the IPM optimization problem to be able to cope with non-realizable gPC coefficients. Consequently, standard filters can be applied to the regularized IPM method. We demonstrate numerical results for the two strategies by investigating the Euler equations with uncertain shock structures in one-and two-dimensional spatial settings. We are able to show a significant reduction of spurious oscillations by the proposed filters.

show abstract

Stable and Accurate Filtering Procedures

Cited by 13 publications

References 10 publications

Stable Dynamical Adaptive Mesh Refinement

Stable Dynamical Adaptive Mesh Refinement

Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

A realizable filtered intrusive polynomial moment method

Contact Info

Product

Resources

About