Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Almquist, Martin; Wang, Siyang; Werpers, Jonatan

doi:10.1137/18m1191609

Cited by 26 publications

(36 citation statements)

References 31 publications

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“…The second choice of SATs uses four penalty terms [28], which has a better stability property for problems with curved interfaces. The method was improved further in [1] from the accuracy perspective when non-periodic boundary conditions are used in the x-direction. In addition, the penalty parameters in [1] are optimized and are sharper than those in [28].…”

Section: The Sbp-sat Methodsmentioning

confidence: 99%

“…The method was improved further in [1] from the accuracy perspective when non-periodic boundary conditions are used in the x-direction. In addition, the penalty parameters in [1] are optimized and are sharper than those in [28]. As will be seen in the numerical experiments, the sharper penalty parameters lead to an improved CFL condition.…”

Section: The Sbp-sat Methodsmentioning

confidence: 99%

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Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Wang¹,

Petersson²

2019

SIAM J. Sci. Comput.

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We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

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Section: The Sbp-sat Methodsmentioning

confidence: 99%

Section: The Sbp-sat Methodsmentioning

confidence: 99%

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Wang¹,

Petersson²

2019

SIAM J. Sci. Comput.

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“…Proof of Theorem 2. 1 We aim to show that Q Q −1 = I , where I is the (n + 1) × (n + 1) identity matrix. Using Q from ( 7) and Q −1 from ( 9), we compute 5) is a consistent difference operator.…”

Section: Theorem 21mentioning

confidence: 99%

“…where we need to show that d dt v 2 H ≤ 0. We will determine the stability limits of σ L,R and τ L,R using a procedure sometimes called the borrowing technique [1,2,7,15,24,30,32]. The idea is to "borrow" a maximum amount γ of "positivity" from A, more precisely as…”

Section: Stabilitymentioning

confidence: 99%

Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative

Eriksson

2021

J Sci Comput

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The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green’s functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.

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“…These ideas can contribute to the creation of a free-energy stable scheme as the one developed in this work. The foundation for the establishment of this family of methods has been laid in [25][26][27][28]. One of the first works is the one presented by Carpenter et al [29] which makes a first attempt to create an entropy-stable scheme using the local discontinuous Galerkin method.…”

Section: Introductionmentioning

confidence: 99%

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

Ntoukas

Manzanero

Rubio

et al. 2021

Journal of Computational Physics

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A novel free-energy stable discontinuous Galerkin method is developed for the Cahn-Hilliard equation with non-conforming elements. This work focuses on dynamic polynomial adaption (p-refinement) and constitutes an extension of the method developed by Manzanero et al. in Journal of Computational Physics 403:109072, 2020, which makes use of the summation-by-parts simultaneous-approximation term technique along with Gauss-Lobatto points and the Bassi-Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates nonconforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free-energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaption is introduced based on the knowledge of the location of the interface between phases. The adaption methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We solve a steady one-dimensional interface test case to initially examine the accuracy of the adaption. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free-energy are recovered with a 35% reduction of the degrees of freedom for the two-dimensional case considered and a 48% reduction for the three-dimensional case.

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Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Cited by 26 publications

References 31 publications

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

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