Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks
Most high-order Euler-type methods have been proposed to solve one-dimensional scalar hyperbolic conservational law. These methods resolve smooth variations in flow parameters accurately and simultaneously identify the discontinuities. A disadvantage of Euler-type methods is the parameter change str...
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2025-05-01
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| author | Valeriy Nikonov |
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| description | Most high-order Euler-type methods have been proposed to solve one-dimensional scalar hyperbolic conservational law. These methods resolve smooth variations in flow parameters accurately and simultaneously identify the discontinuities. A disadvantage of Euler-type methods is the parameter change stretching in the shock over a few mesh cells. In reality, in the shock, the flow properties change abruptly at once for the computational mesh. In our considerations, the mean free path of a flow particle is much smaller than the mesh cell size. This paper describes a modification of the semi-Lagrangian Godunov-type method, which was proposed by the author in the previously published paper. The modified method also does not have numerical viscosity for shocks. In the previous article, a linear law for the distribution of flow parameters was employed for a rarefaction wave when modeling the Shu-Osher problem with the aim of reducing parasitic oscillations. Additionally, the nonlinear law derived from the Riemann invariants was used for the remaining test problems. This article proposes an advanced method, namely, a unified formula for the density distribution of rarefaction waves and modification of the scheme for modeling moderately strong shock waves. The obtained results of numerical analysis, including the standard problem of Sod, the Riemann problem of Lax, the Shu–Osher shock-tube problem and a few author’s test cases are compared with the exact solution, the data of the previous method and the Total Variation Deminishing (TVD) scheme results. This article delineates the further advancement of the numerical scheme of the proposed method, specifically presenting a unified mathematical formulation for an expanded set of test problems. |
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| publishDate | 2025-05-01 |
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| spelling | doaj-art-b51b093e59b64f8d9e147546265fc05b2025-08-20T02:33:55ZengMDPI AGFluids2311-55212025-05-0110513310.3390/fluids10050133Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for ShocksValeriy Nikonov0Aircraft Construction and Design Department, Samara University, Moskovskoye shosse, 34, 443086 Samara, RussiaMost high-order Euler-type methods have been proposed to solve one-dimensional scalar hyperbolic conservational law. These methods resolve smooth variations in flow parameters accurately and simultaneously identify the discontinuities. A disadvantage of Euler-type methods is the parameter change stretching in the shock over a few mesh cells. In reality, in the shock, the flow properties change abruptly at once for the computational mesh. In our considerations, the mean free path of a flow particle is much smaller than the mesh cell size. This paper describes a modification of the semi-Lagrangian Godunov-type method, which was proposed by the author in the previously published paper. The modified method also does not have numerical viscosity for shocks. In the previous article, a linear law for the distribution of flow parameters was employed for a rarefaction wave when modeling the Shu-Osher problem with the aim of reducing parasitic oscillations. Additionally, the nonlinear law derived from the Riemann invariants was used for the remaining test problems. This article proposes an advanced method, namely, a unified formula for the density distribution of rarefaction waves and modification of the scheme for modeling moderately strong shock waves. The obtained results of numerical analysis, including the standard problem of Sod, the Riemann problem of Lax, the Shu–Osher shock-tube problem and a few author’s test cases are compared with the exact solution, the data of the previous method and the Total Variation Deminishing (TVD) scheme results. This article delineates the further advancement of the numerical scheme of the proposed method, specifically presenting a unified mathematical formulation for an expanded set of test problems.https://www.mdpi.com/2311-5521/10/5/133gasshock waveRiemann problemGodunov methodLagrange-type methodnumerical viscosity |
| spellingShingle | Valeriy Nikonov Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks Fluids gas shock wave Riemann problem Godunov method Lagrange-type method numerical viscosity |
| title | Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks |
| title_full | Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks |
| title_fullStr | Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks |
| title_full_unstemmed | Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks |
| title_short | Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks |
| title_sort | modified semi lagrangian godunov type method without numerical viscosity for shocks |
| topic | gas shock wave Riemann problem Godunov method Lagrange-type method numerical viscosity |
| url | https://www.mdpi.com/2311-5521/10/5/133 |
| work_keys_str_mv | AT valeriynikonov modifiedsemilagrangiangodunovtypemethodwithoutnumericalviscosityforshocks |