Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains

Nikkar, Samira; Nordström, Jan

doi:10.1007/978-3-319-19800-2_35

Cited by 12 publications

(36 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With the recently derived interface treatment, it can be shown that the scheme is strongly stable and mimics the continuous estimate in (15). This was previously done in [13] and we do not repeat those derivations here.…”

Section: Stabilitymentioning

confidence: 75%

“…By substituting (14) in (13) and considering the initial conditions U(0, ξ , η) = f L and V (0, ξ , η) = f R , we arrive at to the non-negative and negative eigenvalues, respectively. At the interface, we have n L = (1, 0) T , n R = (−1, 0) T and therefore C L = A L I and C R = −A R I .…”

Section: Well-posednessmentioning

confidence: 99%

“…We have already, in [13], given a thorough analysis for the weak initial and far-field boundary conditions in terms of conservation and stability. Hence, we only display these two terms in (21) for completeness without defining and resolving them further in this article.…”

Section: The Discrete Problemmentioning

confidence: 99%

“…The focus of the SBP-SAT methodology has been, so far, mostly on time-independent spatial domains with a notable exception being [13].…”

Section: Introductionmentioning

confidence: 99%

“…In this article, we extend the techniques introduced in [13] for handling timedependent boundaries in a single domain problem, to a multi-domain context with deforming interfaces. The new time-dependent interface formulation is conservative, provably stable and high order accurate.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Nikkar

Nordstrm

2017

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We introduce an interface/coupling procedure for hyperbolic problems posed on time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions. Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

show abstract

Section: Stabilitymentioning

confidence: 75%

Section: Well-posednessmentioning

confidence: 99%

Section: The Discrete Problemmentioning

confidence: 99%

“…The focus of the SBP-SAT methodology has been, so far, mostly on time-independent spatial domains with a notable exception being [13].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Nikkar

Nordstrm

2017

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

High order weight‐adjusted discontinuous Galerkin methods for wave propagation on moving curved meshes

Guo

Chan

2021

Numerical Meth Engineering

View full text Add to dashboard Cite

This article presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian–Eulerian formulation to map the wave equation from a time‐dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time‐stepping methods. We avoid this step by utilizing an easily invertible weight‐adjusted approximation. The resulting semi‐discrete weight‐adjusted DG scheme is provably energy stable up to a term that (for a fixed time interval) converges to zero with the same rate as the optimal L2 error estimate. Numerical experiments using both polynomial and B‐spline bases verify the high order accuracy and energy stability of proposed methods.

show abstract

A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes

et al. 2016

View full text Add to dashboard Cite

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results.

show abstract

Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains

Cited by 12 publications

References 13 publications

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

High order weight‐adjusted discontinuous Galerkin methods for wave propagation on moving curved meshes

A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes

Contact Info

Product

Resources

About