A Class of Positive Semi-discrete Lagrangian-Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Author(s)

Abreu, Eduardo ^[1] ; Francois, Jean ^[1] ; Lambert, Wanderson ^[2] ; Perez, John ^[3]

Affiliation

^[1] Univ Estadual Campinas, IMECC UNICAMP, Campinas, SP - Brazil

^[2] Univ Fed Alfenas, UNIFAL, Pocos De Caldas, MG - Brazil

^[3] Inst Tecnol Metropolitano, Antioquia - Colombia

• Document type: Journal article
• Source: JOURNAL OF SCIENTIFIC COMPUTING; v. 90, n. 1 JAN 2022.
• Web of Science Citations: 0

Abstract

In this paper, we design and analyze a new class of positive Semi-Discrete Lagrangian-Eulerian (SDLE) schemes for solving multidimensional initial value problems for scalar models and systems of conservation laws. The construction of the schemes is based on the space-time no flow surface region, previously presented and analyzed by the authors for fully-discrete schemes. The implementation of the scheme in the case of systems is a straightforward componentwise application of the multidimensional scalar case, but, importantly, the semi-discrete approach does not require dimensional splitting strategies. Entropy-convergence proof for the multidimensional scalar case is provided via weak asymptotic analysis. We also prove that the new two-dimensional Lagrangian-Eulerian scheme satisfies the scalar maximum principle along with relevant estimates, which also implies the uniqueness of the weak solution satisfying Kruzhkov entropy condition. Moreover, we show that the new semi-discrete Lagrangian-Eulerian scheme, in the more general context of multidimensional hyperbolic systems of conservation laws, also satisfies the positivity principle. Indeed, by using the no flow properties, it is not necessary to obtain the eigenvalues associated with the hyperbolic flux to guarantee the positivity of our numerical scheme. We also use the no flow estimates for numerically stable computations-for scalar equations and multidimensional systems-in a similar fashion to that of the well-known stability condition by Courant-Friedrichs-Lewy (CFL), but without the need to employ the eigenvalues (exact and approximate values) of the relevant Jacobian of the numerical flux functions. Another interesting feature of the no flow Lagrangian-Eulerian construction is that the matrices are symmetric for free (actually, they are diagonal), which is independent for a general class of hyperbolic flux for scalar problems and systems as well. We provide robust numerical examples to verify the theory and illustrate the capabilities of the semi-discrete approach in three cases (1) a 4 x 4 system of compressible Euler flows (Double Mach reflection and Wind tunnel problems), (2) a 3 x 3 shallow-water system of equations with and without bottom discontinuous topography, and (3) 2 x 2 nonstrictly hyperbolic three-phase flows with a resonance point. We also demonstrate the application of our semi-discrete scheme to nontrivial prototype two-dimensional scalar problems that display intricate wave interactions (e.g., 2D inviscid Burgers' equation for an oblique Riemann problem, Buckley-Leverett equation with gravity, and nonlinear equation with non-convex fluxes). We present satisfactory qualitative results for the Orszag-Tang vortex system in a prototype ideal magnetohydrodynamic model with no need of an additional constrained-transport enforcement to preserve the magnetic field. We also verify non-oscillatory and sufficiently accurate numerical solutions for the 2D Euler system, isentropic smooth vortex convection problem. The multidimensional SDLE scheme retains simplicity, with a very good resolution, is efficient in terms of computational and memory cost, and is simple to implement as well since no (local) Riemann problems are solved; hence, time-consuming field-by-field type decompositions are avoided in the case of systems. These features are significant and ensure the simplicity and power of this class of positive semi-discrete schemes. (AU)

• FAPESP's process: 19/20991-8 - Study of partial differential algebraic equations of hyperbolic-parabolic dominance with relaxation: theory, numerics and applications
• Grantee: Eduardo Cardoso de Abreu
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil