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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 × 4 system of compressible Euler flows (Double Mach reflection and Wind tunnel problems), (2) a 3 × 3 shallow-water system of equations with and without bottom discontinuous topography, and (3) 2 × 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. [ABSTRACT FROM AUTHOR]

We study systems of partial differential-algebraic equations (PDAEs) of first order. Classical solutions of the theory of hyperbolic partial differential equation such as discontinuities (shock and contact discontinuities), rarefactions and diffusive traveling waves are extended for variables living on a surface S , which is defined as solution of a set of algebraic equations. We propose here an alternative formulation to study numerically and theoretically the PDAEs by changing the algebraic conditions into partial differential equations with relaxation source terms (PDREs). The solution of such relaxed systems is proved to tend to the surface S , i.e., to satisfy the algebraic equations for long times. We formulate a unified numerical scheme for systems of PDAEs and PDREs. This scheme is naturally parallelizable and has faster convergence. We do not perform a rigorous analysis about the convergence or accuracy for the method, the evidence of its effectiveness is presented by means of simulations for physical and synthetical problems. [ABSTRACT FROM AUTHOR]

Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a twofold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. Using the projection method we show the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid finite element method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling fundamental two-phase flows in porous media. [ABSTRACT FROM AUTHOR]

We developed an unsplitting finite volume scheme to account the delicate nonlinear balance between numerical approximations of the hyperbolic flux function and the source linked to balance laws. The method is Riemann-solver-free and no upwinding technique is used. By means of this new approach, we conducted an analysis for two new models of balance laws linked to compositional and thermal flow in porous media problems, under and without a thermodynamic equilibrium hypothesis. For concreteness, we adopt the nitrogen and steam injection models in a porous media. To this model we found an interesting behavior linked to the relaxation term, which is the existence of a non-monotonic traveling wave. We applied this numerical technique to others well-known differential models with relaxation terms available in the literature. Qualitatively we were able to reproduce the expected results. [ABSTRACT FROM AUTHOR]

In this work, we design a new class of fully discrete Lagrangian–Eulerian schemes on triangular grids to approximate nonlinear multidimensional initial value problems for scalar models and multidimensional systems of conservation. The numerical approach is based on the improved concept of multidimensional no-flow curves. We also provide a convergence proof towards the entropy solution to the scalar problem ut(t,x)+\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u_{t}(t,{\textbf{x}})+$$\end{document}div(f(u(t,x)))=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ ({\textbf{f}}(u(t,{\textbf{x}})))=0$$\end{document}, with u0∈L∞(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u_{0} \in L^{\infty }({\mathbb {R}}^{2})$$\end{document}, which is analyzed in the effective setting of the properties of uniqueness and regularity of the entropy process solutions in the Lt∞(Lx1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_{t}^{\infty }(L_{{\textbf{x}}}^{1})$$\end{document}-norm and linked to the entropy measure-valued solutions. Additionally, we obtain optimal a priori error estimates as O(h12)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}(h^{\frac{1}{2}})$$\end{document} on equilateral triangular meshes. In the general context of multidimensional hyperbolic systems of conservation laws, we show that the Lagrangian–Eulerian scheme on triangular grids also satisfies the positivity principle proposed by Lax and Liu (J Comput Phys 5(2):133–156, 1996; J Comput Phys 187:428–440, 2003) and a type of weak positivity principle. We found a connection (given in the “Appendix”) between the notion of no-flow curves (viewed as a vector field with locally bounded ΓM-variation\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma ^{M}{-}variation$$\end{document}) and the results of Bressan in the context of (local) existence and continuous dependence for discontinuous O.D.E.’s as introduced in Bressan (Proc Am Math Soc 104(3):772–778, 1988) and Bressan and Colombo (Boll Un Mat Ital 4(2):295–311, 1990). We present numerical solutions for nontrivial 2D hyperbolic problems, e.g., 4 by 4 compressible Euler equations (Double Mach Reflection problem and Mach 3 wind tunnel flow), a nonclassical 2 by 2 three-phase flow system of nonstrictly hyperbolic conservation laws (with a resonance/umbilic point), and the 3 by 3 shallow-water system (with and without bottom topography), and also 8 by 8 Orszag-Tang vortex system in magnetohydrodynamics. [ABSTRACT FROM AUTHOR]