# New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. New US-WENO Lagrangian Scheme

#### 2.1. Two-Dimensional Case

#### 2.1.1. Finite Volume Discretization

#### 2.1.2. Unequal-Sized WENO Reconstruction

#### 2.1.3. The Velocity of Vertex

#### 2.1.4. Time Discretization

#### 2.2. Three-Dimensional Case

#### 2.2.1. Finite Volume Discretization

#### 2.2.2. Unequal-Sized WENO Reconstruction

#### 2.2.3. The Velocity of Vertex

#### 2.2.4. Time Discretization

## 3. Numerical Results

#### 3.1. Accuracy Test

**Example 1.**

**Example 2.**

#### 3.2. Two-Dimensional Lagrangian Tests

**Example 3.**

**Example 4.**

**Example 5.**

**Example 6.**

**Example 7.**

**Example 8.**

#### 3.3. Two-Dimensional ALE Tests

**Example 9.**

**Example 10.**

#### 3.4. Three-Dimensional Lagrangian Tests

**Example 11.**

**Example 12.**

**Example 13.**

## 4. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Cheng, J.; Shu, C.-W. A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. J. Comput. Phys.
**2007**, 227, 1567–1596. [Google Scholar] [CrossRef] - Cheng, J.; Shu, C.-W. A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations. Commun. Comput. Phys.
**2008**, 4, 1008–1024. [Google Scholar] - Hu, C.; Shu, C.-W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys.
**1999**, 150, 97–127. [Google Scholar] [CrossRef] - Zhang, Y.T.; Shu, C.-W. Third order WENO scheme on three dimensional tetrahedral meshes. Commun. Comput. Phys.
**2009**, 5, 836–848. [Google Scholar] - Luo, H.; Baum, J.D.; Löhner, R. On the computation of multi-material flows using ALE formulation. J. Comput. Phys.
**2004**, 194, 304–328. [Google Scholar] [CrossRef] - Barlow, A.J.; Maire, P.-H.; Rider, W.J.; Rieben, R.N.; Shashkov, M.J. Arbitrary Lagrangian Eulerian methods for modeling high-speed compressible multimaterial flows. J. Comput. Phys.
**2016**, 332, 603–665. [Google Scholar] [CrossRef] - Burton, D.E. Exact Conservation of Energy and Momentum in Staggered-Grid Hydrodynamics with Arbitrary Connectivity, Advances in the Free Lagrange Method; Springer: New York, NY, USA, 1990. [Google Scholar]
- Burton, D.E. Multidimensional Discretization of Conservation Laws for Unstructured Polyhedral Grids; Technical Report UCRL-JC-118306; Lawrence Livermore National Laboratory: Livermore, CA, USA, 1990.
- Caramana, E.J.; Rousculp, C.L.; Burton, D.E. A compatible, energy and symmetry preserving lagrangian hydrodynamics algorithm in three-dimensional Cartesian geometry. J. Comput. Phys.
**2000**, 157, 89–119. [Google Scholar] [CrossRef] - Caramana, E.J.; Shashkov, M.J.; Whalen, P.P. Formulations of artificial viscosity for multi-dimensional shock wave computations. J. Comput. Phys.
**1998**, 144, 70–97. [Google Scholar] [CrossRef] - Loubère, R.; Maire, P.-H.; Vachal, P. 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity. Int. J. Numer. Methods Fluids
**2013**, 72, 22–42. [Google Scholar] [CrossRef] - Cheng, J.; Shu, C.-W. Positivity-preserving Lagrangian scheme for multi-material compressible flow. J. Comput. Phys.
**2014**, 257, 143–168. [Google Scholar] [CrossRef] - Georges, G.; Breil, J.; Maire, P.-H. A 3D GCL compatible cell-centered Lagrangian scheme for solving gas dynamics equations. J. Comput. Phys.
**2016**, 305, 921–941. [Google Scholar] [CrossRef] - Liu, W.; Cheng, J.; Shu, C.-W. High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations. J. Comput. Phys.
**2009**, 228, 8872–8891. [Google Scholar] [CrossRef] - Maire, P.-H. A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry. J. Comput. Phys.
**2009**, 228, 6882–6915. [Google Scholar] [CrossRef] - Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J. A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput.
**2007**, 29, 1781–1824. [Google Scholar] [CrossRef] - Maire, P.-H.; Breil, J. A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems. Int. J. Numer. Meth. Fluids
**2008**, 56, 1417–1423. [Google Scholar] [CrossRef] - Maire, P.-H.; Nkonga, B. Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. J. Comput. Phys.
**2009**, 228, 799–821. [Google Scholar] [CrossRef] - Xu, X.; Dai, Z.H.; Gao, Z.M. A 3D cell-centered Lagrangian scheme for the ideal magnetohydrodynamics equations on unstructured meshes. Comput. Methods Appl. Mech. Eng.
**2018**, 342, 490–508. [Google Scholar] [CrossRef] - Munz, C.D. On Godunov-type schemes for Lagrangian gas dynamics. SIAM J. Numer. Anal.
**1994**, 31, 17–42. [Google Scholar] [CrossRef] - Després, B.; Mazeran, C. Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers. Comptes Rendus Méc.
**2003**, 331, 475–480. [Google Scholar] [CrossRef] - Després, B.; Mazeran, C. Lagrangian gas dynamics in two-dimensions and Lagrangian systems. Arch. Ration. Mech. Anal.
**2005**, 178, 327–372. [Google Scholar] [CrossRef] - Harten, A.; Engquist, B.; Osher, S.; Chakravathy, S. Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys.
**1987**, 71, 231–303. [Google Scholar] [CrossRef] - Harten, A.; Osher, S. Uniformly high-order accurate non-oscillatory schemes I. SIAM J. Numer. Anal.
**1987**, 24, 279–309. [Google Scholar] [CrossRef] - Jiang, G.; Shu, C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys.
**1996**, 126, 202–228. [Google Scholar] [CrossRef] - Liu, X.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys.
**1994**, 115, 200–212. [Google Scholar] [CrossRef] - Titarev, V.A.; Toro, E.F. Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys.
**2004**, 201, 238–260. [Google Scholar] [CrossRef] - Boscheri, W.; Balsara, D.S.; Dumbser, M. Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. J. Comput. Phys.
**2014**, 267, 112–138. [Google Scholar] [CrossRef] - Boscheri, W.; Dumbser, M. Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun. Comput. Phys.
**2013**, 14, 1174–1206. [Google Scholar] [CrossRef] - Boscheri, W.; Dumbser, M. A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. J. Comput. Phys.
**2014**, 275, 484–523. [Google Scholar] [CrossRef] - Boscheri, W.; Dumbser, M.; Balsara, D.S. High order Lagrangian ADER-WENO schemes on unstructured meshes—Application of several node solvers to hydrodynamics and magnetohydrodynamics. Int. J. Numer. Methods Fluids
**2014**, 76, 737–778. [Google Scholar] [CrossRef] - Dumbser, M. Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng.
**2014**, 280, 57–83. [Google Scholar] [CrossRef] - Dumbser, M.; Käser, M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys.
**2007**, 221, 693–723. [Google Scholar] [CrossRef] - Dumbser, M.; Käser, M.; Titarev, V.A.; Toro, E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys.
**2007**, 226, 204–243. [Google Scholar] [CrossRef] - Anderson, R.W.; Dobrev, V.A.; Kolev, T.V.; Rieben, R.N.; Tomov, V.Z. High-order multi-material ale hydrodynamics. SIAM J. Sci. Comput.
**2018**, 40, B32–B58. [Google Scholar] [CrossRef] - Boscheri, W.; Balsara, D.S. High order direct Arbitrary-Lagrangian-Eulerian (ALE) P
_{N}P_{M}schemes with WENO Adaptive-Order reconstruction on unstructured meshes. J. Comput. Phys.**2019**, 398, 108899. [Google Scholar] [CrossRef] - Dobrev, V.; Kolev, T.; Rieben, R. High-order curvilinear finite element methods for Lagrangian hydrodynamics. SIAM J. Sci. Comput.
**2012**, 34, B606–B641. [Google Scholar] [CrossRef] - Gaburro, E. A unified framework for the solution of hyperbolic PDE systems using high order direct Arbitrary-Lagrangian-Eulerian schemes on moving unstructured meshes with topology change. Arch. Comput. Methods Eng.
**2020**, 28, 1–73. [Google Scholar] [CrossRef] - Gaburro, E.; Boscheri, W.; Chiocchetti, S.; Klingenberg, C.; Springel, V.; Dumbser, M. High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes. J. Comput. Phys.
**2020**, 407, 109167. [Google Scholar] [CrossRef] - Lei, N.; Cheng, J.; Shu, C.-W. A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes. Comput. Methods Appl. Mech. Eng.
**2021**, 373, 113497. [Google Scholar] [CrossRef] - Lei, N.; Cheng, J.; Shu, C.-W. A high order positivity-preserving conservative WENO remapping method on 3D tetrahedral meshes. Comput. Methods Appl. Mech. Eng.
**2022**, 395, 115037. [Google Scholar] [CrossRef] - Pan, L.; Xu, K. An arbitrary-Lagrangian-Eulerian high-order gas-kinetic scheme for three-dimensional computations. J. Sci. Comput.
**2021**, 88, 1–29. [Google Scholar] [CrossRef] - Pan, L.; Zhao, F.X.; Xu, K. High-order ALE gas-kinetic scheme with WENO reconstruction. J. Comput. Phys.
**2020**, 417, 109558. [Google Scholar] [CrossRef] - Shi, J.; Hu, C.; Shu, C.-W. A technique of treating negative weights in WENO schemes. J. Comput. Phys.
**2002**, 175, 108–127. [Google Scholar] [CrossRef] - Liu, Y.; Zhang, Y.T. A robust reconstruction for unstructured WENO schemes. J. Sci. Comput.
**2013**, 54, 603–621. [Google Scholar] [CrossRef] - Zhu, J.; Qiu, J. A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys.
**2016**, 318, 110–121. [Google Scholar] [CrossRef] - Zhu, J.; Qiu, J. New finite volume weighted essentially non-oscillatory scheme on triangular meshes. SIAM J. Sci. Comput.
**2018**, 40, 903–928. [Google Scholar] [CrossRef] - Zhu, J.; Qiu, J. A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes. J. Comput. Phys.
**2017**, 349, 220–232. [Google Scholar] [CrossRef] - Batten, P.; Leschziner, M.A.; Goldberg, U.C. Average-state Jacobians and implicit methods for compressible viscous and turbulent flows. J. Comput. Phys.
**1997**, 137, 38–78. [Google Scholar] [CrossRef] - Roe, P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys.
**1981**, 43, 357–372. [Google Scholar] [CrossRef] - Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys.
**1988**, 77, 439–471. [Google Scholar] [CrossRef] - Qiu, J.; Shu, C.-W. On the construction, comparison, and local characteristic decomposition for high order central WENO schemes. J. Comput. Phys.
**2002**, 183, 187–209. [Google Scholar] [CrossRef] - Sod, G. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys.
**1978**, 27, 1–31. [Google Scholar] [CrossRef] - Lax, P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math.
**1954**, 7, 159–193. [Google Scholar] [CrossRef] - Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys.
**1989**, 83, 32–78. [Google Scholar] [CrossRef] - Woodward, P.; Colella, P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys.
**1984**, 54, 115–173. [Google Scholar] [CrossRef] - Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd ed.; Springer: New York, NY, USA, 2009. [Google Scholar]
- Dukowicz, J.K.; Meltz, B. Vorticity errors in multidimensional Lagrangian codes. J. Comput. Phys.
**1992**, 99, 115–134. [Google Scholar] [CrossRef] - Kamm, J.R.; Timmes, F.X. On Efficient Generation of Numerically Robust Sedov Solutions; Technical Report LA-UR-07-2849; Los Alamos National Laboratory: Los Alamos, NM, USA, 2007.
- Cheng, J.; Shu, C.-W. A high order accurate conservative remapping method on staggered meshes. Appl. Numer. Math.
**2008**, 58, 1042–1060. [Google Scholar] [CrossRef] - Tang, H.Z.; Tang, T. Moving mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal.
**2003**, 41, 487–515. [Google Scholar] [CrossRef] - Schulz-Rinne, C.W.; Collins, J.P.; Glaz, H.M. Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput.
**1993**, 14, 1394–1414. [Google Scholar] [CrossRef] - Galera, S.; Maire, P.-H.; Breil, J. A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction. J. Comput. Phys.
**2010**, 229, 5755–5787. [Google Scholar] [CrossRef] - Kucharik, M.; Garimella, R.V.; Schofield, S.P.; Shashkov, M.J. A comparative study of interface reconstruction methods for multi-material ALE simulations. J. Comput. Phys.
**2010**, 229, 2432–2452. [Google Scholar] [CrossRef] - Morgan, N.R.; Lipnikov, K.N.; Burton, D.E.; Kenamond, M.A. A Lagrangian staggered grid Godunov-like approach for hydrodynamics. J. Comput. Phys.
**2014**, 259, 568–597. [Google Scholar] [CrossRef]

**Figure 3.**Sod and Lax problem. Density cut at $y=0$. Solid line: the exact solution; squares: the results of US-WENO Lagrangian scheme.

**Figure 4.**Shu–Osher problem. T = 1.8. Density cut at $y=0$. Solid line: the reference solution; squares: the results of US-WENO Lagrangian scheme. (

**Left**): overall view. (

**Right**): enlarged view.

**Figure 5.**Blast wave problem. T = 0.038. Density cut at $y=0$. Solid line: the reference solution; squares: the results of US-WENO Lagrangian scheme. (

**Left**): overall view. (

**Right**): enlarged view.

**Figure 6.**The 2D Sedov problem. T = 1. (

**Top left**): mesh and density distributions; (

**top right**): mesh and pressure distributions; (

**bottom**): density in diagonal cells with respect to the radius.

**Figure 7.**The 2D Sod problem. T = 0.2. (

**Top left**): mesh distributions; (

**top right**): density distributions; (

**bottom left**): pressure distributions; (

**bottom right**): density profiles along $y=0$.

**Figure 9.**The 2D Saltzman problem. T = 0.6. (

**Top left**): mesh and density distributions; (

**top right**): mesh and pressure distributions; (

**bottom left**): density profiles along $y=0$ comparison with analytical solution; (

**bottom right**): pressure profiles along $y=0$ comparison with analytical solution.

**Figure 12.**The 3D Sedov problem. T = 1. (

**Top left**): mesh and density distributions; (

**top right**): mesh and pressure distributions; (

**bottom**): comparison between analytical and numerical density profiles along the diagonal cell.

**Figure 13.**The 3D Sod problem. T = 0.25. (

**Top left**): mesh and density distributions; (

**top right**): mesh and pressure distributions; (

**bottom left**): comparison between analytical and numerical density profiles along the diagonal cell; (

**bottom right**): comparison between analytical and numerical pressure profiles along the diagonal cell.

**Figure 15.**The 3D Saltzman problem. T = 0.6. (

**Top left**): mesh and density distributions; (

**top right**): mesh and pressure distributions; (

**bottom left**): density profiles plotted against the analytical solution; (

**bottom right**): pressure profiles plotted against the analytical solution.

Mesh | ${\mathit{L}}^{1}$ Error | Order | ${\mathit{L}}^{\mathit{\infty}}$ Error | Order | ${\mathit{L}}^{1}$ Error | Order | ${\mathit{L}}^{\mathit{\infty}}$ Error | Order |
---|---|---|---|---|---|---|---|---|

Linear weights (1) | Linear weights (2) | |||||||

$20\times 20$ | 4.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | ||||

$40\times 40$ | 6.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.65 | 1.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 | 6.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.65 | 1.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 |

$60\times 60$ | 1.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.91 | 5.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.86 | 1.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.91 | 5.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.86 |

$80\times 80$ | 8.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.97 | 2.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.95 | 8.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.97 | 2.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.95 |

$100\times 100$ | 4.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.98 | 1.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.97 | 4.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.98 | 1.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.97 |

Linear weights (3) | Linear weights (4) | |||||||

$20\times 20$ | 4.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | ||||

$40\times 40$ | 6.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.65 | 1.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 | 6.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.65 | 1.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 |

$60\times 60$ | 1.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.91 | 5.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.86 | 1.98 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.91 | 5.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.86 |

$80\times 80$ | 8.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.97 | 2.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.95 | 8.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.97 | 2.54 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.95 |

$100\times 100$ | 4.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.98 | 1.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.97 | 4.33 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.98 | 1.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.97 |

Mesh | ${\mathit{L}}^{1}$ Error | Order | ${\mathit{L}}^{\mathit{\infty}}$ Error | Order | ${\mathit{L}}^{1}$ Error | Order | ${\mathit{L}}^{\mathit{\infty}}$ Error | Order |
---|---|---|---|---|---|---|---|---|

Linear weights (1) | Linear weights (2) | |||||||

$10\times 10\times 10$ | 8.25 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.62 × 10^{−3} | 8.25 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.62 × 10^{−3} | ||||

$20\times 20\times 20$ | 8.52 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.28 | 1.99 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.03 | 8.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.27 | 2.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.02 |

$30\times 30\times 30$ | 2.38 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.15 | 6.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.94 | 2.38 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.15 | 6.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 |

$40\times 40\times 40$ | 1.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.84 | 2.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.14 | 1.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.84 | 2.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.14 |

$50\times 50\times 50$ | 5.23 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.12 | 1.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 | 5.23 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.12 | 1.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 |

Linear weights (3) | Linear weights (4) | |||||||

$10\times 10\times 10$ | 8.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.62 × 10^{−3} | 8.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.62 × 10^{−3} | ||||

$20\times 20\times 20$ | 8.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.28 | 2.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.02 | 8.53 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.28 | 2.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.02 |

$30\times 30\times 30$ | 2.38 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.15 | 6.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 | 2.38 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.15 | 6.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 |

$40\times 40\times 40$ | 1.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.84 | 2.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.14 | 1.05 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 2.84 | 2.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.14 |

$50\times 50\times 50$ | 5.23 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.12 | 1.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 | 5.23 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 3.12 | 1.26 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.96 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tan, Y.; Lv, H.; Zhu, J.
New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations. *Mathematics* **2023**, *11*, 4842.
https://doi.org/10.3390/math11234842

**AMA Style**

Tan Y, Lv H, Zhu J.
New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations. *Mathematics*. 2023; 11(23):4842.
https://doi.org/10.3390/math11234842

**Chicago/Turabian Style**

Tan, Yan, Hui Lv, and Jun Zhu.
2023. "New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations" *Mathematics* 11, no. 23: 4842.
https://doi.org/10.3390/math11234842