Combined DG Scheme That Maintains Increased Accuracy in Shock Wave Areas

Ladonkina, M. E.; Nekliudova, O. A.; Остапенко, В. В.; Тишкин, В. Ф.

doi:10.1134/s106456241906005x

Cited by 22 publications

(4 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The key idea in the construction of the combined schemes (see, e.g., [16,21,35]) is to have two copies of the computed solutions: one of them is the RBM solution, which is computed throughout the entire computational domain and which is highly accurate in the smooth areas, and the second one is the non-oscillatory solution computed with the help of either the CU or A-WENO scheme near the shock regions only.…”

Section: Combined Schemesmentioning

confidence: 99%

“…Relying on the conducted convergence study, we develop two new combined schemes based on the RBM as a basic scheme and either the CU or A-WENO as an internal scheme. The technique for constructing combined schemes has been recently proposed in [16,21,35]. In combined schemes, the numerical solution is first constructed in the entire computational domain according to the basic nonmonotone scheme that maintains increased accuracy inside the shock influence areas.…”

Section: Introductionmentioning

confidence: 99%

“…A major novelty and advantage of the new combined schemes is in the way the "rough" parts of the computed solution are detected. Unlike the combined schemes from [16,21,35], where the smoothness was determined using the size of the solution gradients, here we use the method, which is based on the weak local residual (WLR) introduced in [11,12]. Using the fact that the magnitudes of the WLR in the smooth areas and near the shocks are significantly different, we can accurately determine "rough" areas, and this leads to a very good agreement between the numerical solutions obtained by the basic and internal schemes at the boundary of the internal computational domain.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes

Chu

Ковыркина

Kurganov

et al. 2023

Numerical Methods Partial

View full text Add to dashboard Cite

We study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.

show abstract

Section: Combined Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes

Chu

Ковыркина

Kurganov

et al. 2023

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…Elastic wave propagation is most frequently modeled by applying modifications of finite-difference methods [3,4], spectral element methods [5,6], and discontinuous Galerkin methods [7,8]. Elastic waves in geological media can also be successfully modeled using the grid-characteristic method [9,10].…”

mentioning

confidence: 99%

Joint Modeling of Wave Phenomena by Applying the Grid-Characteristic Method and the Discontinuous Galerkin Method

Petrov

Favorskaya²

2022

Dokl. Math.

View full text Add to dashboard Cite

The aim of this work is to develop a hybrid computational method that combines the grid-characteristic method on regular structured grids with the discontinuous Galerkin method on unstructured tetrahedral grids. The proposed method makes it possible to describe integration domains with complex-shaped boundaries and contact boundaries and to calculate seismic fields taking into account the topography of the Earth’s surface, while saving computational resources. This modification of the method in the three-dimensional case has been proposed in this paper for the first time. Examples of using the developed method for calculating elastic wave phenomena arising during seismic prospecting are given. By way of testing, a comparison is made with results produced by the grid-characteristic method on curvilinear structured grids. The proposed hybrid method can be used not only for seismic prospecting, but also for calculating wave phenomena in other objects of complex shape.

show abstract