Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Tetrahedral Meshes for Conservative and Nonconservative Hyperbolic Systems in 3D

1 Introduction

• Following fluid motion can locate and track material interfaces and contact waves more precisely than classical Eulerian methods on fixed grids.Lagrangian schemes may deform the mesh severely over time, motivating remeshing or rezoning.Source: §1 · pp. 2–3
• Earlier Lagrangian finite-volume schemes discussed here were at most second order in space and time, while subsequent work introduced higher-order spatial reconstruction and time-stepping approaches.Prior developments include ENO reconstruction and ADER-WENO schemes with local space-time Galerkin predictors.Source: §1 · p. 3
• The work extends the algorithm in [13,44] to moving unstructured tetrahedral meshes in 3D, reporting the first better-than-second-order Lagrangian finite-volume method on such meshes.The considered systems are the Euler equations, ideal classical MHD, and the seven-equation Baer-Nunziato model with stiff source terms.Source: §1 · p. 4
• The paper extends to moving unstructured tetrahedral meshes in three dimensions and applies the node solvers proposed by Maire and Cheng and Shu.The ALE approach allows local mesh velocity to be chosen independently from local fluid velocity.Source: §1 · p. 4

2 Numerical method

• The scheme stores and evolves piecewise-constant cell averages on tetrahedral elements, then uses WENO reconstruction for spatial accuracy and a local space-time predictor for temporal accuracy.The governing systems may include non-conservative products and stiff source terms.Source: §2 · pp. 5–6

2.1 Polynomial WENO reconstruction

• The polynomial WENO reconstruction uses nine stencils per tetrahedron—one central, four forward, and four backward—and supports high-order accuracy on unstructured 3D tetrahedral meshes.Stencil elements are added recursively until a prescribed total, which depends on reconstruction degree M, is reached.Source: §2.1 · pp. 6–7
• The positivity-preserving flattener extends to moving unstructured tetrahedral meshes, and a node-based flattener propagates its effect through elements sharing a vertex.The technique is switched off by default and used only for test problems where necessary to reach the final time.Source: §2.1 · pp. 9–11

2.2 Local space-time Discontinuous Galerkin predictor on moving curved tetrahedra

• The element-local space-time DG predictor evolves reconstructed polynomials without neighbor information, producing degree-M space-time polynomials and accommodating stiff source terms on moving 3D tetrahedral meshes.This extends the moving-grid approach to tetrahedral meshes in three dimensions.Source: §2.2 · p. 11
• The local predictor uses space-time Lagrange basis functions and an isoparametric mapping, then solves an element-local nonlinear system for the predicted solution.For stiff algebraic source terms, the source discretization must be implicit.Source: §2.2 · pp. 11, 14, 15

2.3 Mesh motion

• Mesh motion proceeds through Lagrangian, rezoning, and relaxation steps to maintain mesh quality while preserving the benefits of fluid-following motion.Strong mesh deformation can compress, twist, or tangle elements, so rezoning and relaxation are used to improve mesh quality while retaining Lagrangian accuracy where possible.Source: §2.3 · p. 16

2.3.1 The Lagrangian step.

• The authors extend the NScs and NSm node solvers to 3D to assign each vertex a unique velocity from its neighboring elements.NScs uses a mass-weighted average velocity, while NSm derives the node velocity from total-energy conservation and sub-cell forces.Source: §2.3.1 · pp. 16–17

2.3.2 The rezoning step.

• Rezoning improves local mesh quality by optimizing interior-node positions and using a simplified feasible-set method for boundary nodes.The procedure minimizes neighboring-element Jacobian condition numbers for free nodes and computes boundary-node positions as volume-weighted averages of neighboring element barycenters.Source: §2.3.2 · pp. 19–20

2.3.3 The relaxation step.

• The relaxation step aims to keep the ALE mesh as Lagrangian as possible, avoiding rezoning nodes unless necessary for the computation.Source: §2.3.3 · p. 20
• Galera et al.’s node relaxation blends Lagrangian and rezoned positions using deformation-dependent ω_k∈[0,1], with ω_k=0 for rigid motion and ω_k=1 for pure rezoning.Source: §2.3.3 · pp. 20–21

2.4 Finite volume scheme

• For nonconservative systems, a path-conservative jump term resolves predictor discontinuities at space-time faces, with ALE Rusanov- or Osher-type fluxes available.The path integral uses a straight-line segment; the scheme also uses multidimensional Gaussian quadrature for volume and surface integrals.Source: §2.4 · pp. 21, 22, 24, 25
• The one-step ALE finite-volume update integrates balance laws over each tetrahedron’s four-dimensional space-time control volume and sums fluxes across its evolving faces.The update uses the space-time boundary of the moving tetrahedron.Source: §2.4 · pp. 21, 24
• Integration over the closed space-time control volume automatically respects the geometric conservation law.Source: §2.4 · p. 25

• The test problems cover compressible Euler, ideal classical MHD, and Baer-Nunziato multiphase-flow equations, including conservative and non-conservative systems.Source: §3 · p. 26
• Euler and MHD use node solver NSm, while Baer-Nunziato uses NScs to compute mesh velocity.Source: §3 · p. 26
• The flattener is normally disabled and is activated explicitly for selected test cases.Source: §3 · p. 26

3.1.1 Numerical convergence studies

• The density convergence study reports results for first- through sixth-order schemes on successively refined tetrahedral meshes.The error norms are measured at t = 1.0; the mesh size is defined from the maximum circumsphere diameter of final-domain tetrahedra.Source: Table 1 · p. 28, §3.1.1 · p. 28, Figure 5 · p. 28
• Flux evaluation is the most expensive algorithm stage, while WENO reconstruction takes a larger share as order increases.The local space-time predictor is the least expensive stage in the tested second- through fourth-order schemes.Source: §3.1.1 · p. 29, Table 2 · p. 29

3.1.2 The Sod shock tube problem

• The third-order Sod calculation resolves the contact wave with one intermediate point and agrees well with the exact solution for density, pressure, and horizontal velocity.Although one-dimensional, the test is nontrivial on unstructured meshes because faces need not align with the coordinate axes or fluid motion.Source: §3.1.2 · pp. 29–30, Figure 6 · p. 30

3.1.3 Three-dimensional explosion problem

• The fourth-order explosion calculation agrees with the radial reference solution and resolves the contact wave between the spherical shock and rarefaction fan.The reported mesh sequence shows progressive compression of tetrahedra at the shock front.Source: §3.1.3 · p. 31, Figure 7 · p. 32, Figure 8 · p. 33

3.1.4 The Kidder problem

• For the Kidder shell-compression test, the fourth-order scheme compares numerical and analytical internal and external radii, with absolute errors reported at the final time.The computational domain is one eighth of the shell, with the analytical solution prescribed on its inner and outer boundaries.Source: §3.1.4 · pp. 33–34, Table 3 · p. 34, Figure 9 · p. 34

3.1.5 The Saltzman problem

• In the Saltzman piston-driven shock test, third-order density and velocity results agree well with the exact solution at t = 0.6.Density decreases near the piston due to wall heating; positivity preservation smears unphysical oscillations at the shock.Source: §3.1.5 · pp. 34, 36, Figure 10 · p. 35, Figure 11 · p. 36

3.1.6 The Sedov problem

• The third-order Sedov calculation compares density on two meshes with the exact radial solution, whose spherical shock lies at r = 1 with density peak ρ = 6 at t = 1.0.The test uses a positivity-preserving algorithm and the Rusanov-type flux.Source: §3.1.6 · p. 37

3.2 The magnetohydrodynamics (MHD) equations

• The MHD equations require magnetic-field divergence to remain zero; the scheme adds a GLM variable and scalar PDE to transport divergence errors out of the domain.The cleaning speed is denoted c_h.Source: §3.2 · pp. 37, 39

3.2.1 The MHD rotor problem

• The rotor solution qualitatively agrees with prior results, while rezoning keeps the mesh reasonably shaped despite strong velocity-driven deformations.The simulation uses the third-order scheme with a Rusanov-type flux.Source: §3.2.1 · p. 40

3.2.2 The MHD blast wave problem

• The blast-wave test models a strong circular fast magnetosonic shock moving from the center toward the boundaries, with density, pressure, velocity, and magnetic-field results shown at t_f = 0.01.The results use the third-order scheme with the Rusanov-type flux.Source: §3.2.2 · p. 40

3.3 The Baer-Nunziato model of compressible two-phase flows

• The model represents solid and gas phases with volume fractions constrained to sum to one, and uses simplified interphase drag and pressure-relaxation source terms.The interface velocity is chosen as the solid velocity and the interface pressure as the gas pressure.Source: §3.3 · pp. 41–43
• The model’s material interfaces are jumps in volume fraction; for the tests, setting mesh velocity to interface velocity is intended to resolve material contacts properly.Source: §3.3 · pp. 43–44

3.3.1 Riemann problems

• For RP1–RP3, numerical solutions agree very well with reference solutions, and the material contact is well resolved in every case.RP1 and RP2 use the Osher-type method; RP3 uses the more robust Rusanov-type method.Source: §3.3.1 · p. 44

3.3.2 Spherical explosion problems

• Across three spherical explosion problems, the third-order scheme agrees with the one-dimensional reference, and the first-phase contact is very well resolved with mesh velocity set to the interface velocity.The path-conservative Osher-type method is used for lower dissipation and better contact resolution; EP3 uses stiff interphase drag.Source: §3.3.2 · pp. 44, 48

4 Conclusions

• The proposed scheme is a high-order one-step ALE ADER-WENO finite-volume method on unstructured tetrahedral meshes, reported as the first better-than-second-order Lagrangian finite-volume scheme on such meshes.Source: §4 · p. 48
• Convergence studies reach sixth order in space and time, and classical tests cover Euler, MHD, and the seven-equation Baer-Nunziato model, with comparisons to reference solutions where possible.Source: §4 · p. 48
• Future work includes extending the method to PNPM, MOOD limiting, and local time stepping, as well as applying it to more realistic engineering and physics simulations.Source: §4 · p. 52