On the Construction of Combined Finite-Difference Schemes of High Accuracy

Ковыркина, О. А.; Остапенко, В. В.

doi:10.1134/s1064562418010246

Cited by 30 publications

(4 citation statements)

References 9 publications

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“…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%

“…FIGURE16 Example 3: Average experimental rates of pointwise convergence for the CU (left), RBM (middle), and A-WENO (right) schemes.…”

mentioning

confidence: 99%

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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

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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.

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Section: Combined Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%