Systems Of Conservation Laws Research Articles

Well-Balanced Adaptive Compact Approximate Taylor Methods for Systems of Balance Laws

Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Pares in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy, and thus the width of the stencils, is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests.

P-I Boundary Control of Countercurrent Heat Exchanger

Heat exchangers are generally described by two coupled partial differential equations (PDEs), and belong to the class of systems of conservation laws. In this paper we propose a result on the regulation of heat exchangers, in which we are choosing a proportional-integral (PI)-controller implemented on an output feedback from a boundary condition. First, a study on the existence and uniqueness of solutions of uncontrolled system is carried out, and we give a condition on the parameters of the system that ensures that the control-observation problem is well-posedness. Secondly we successively study the stability of the system controlled by a proportional and integral action, when the gains are adjusted following respectively a classic Lyapunov approach, and using a recent result that combines Lyapunov's approach and operator perturbation theory.

A new fifth order finite difference WENO scheme to improve convergence rate at critical points

In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two-dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is applied to construct a new global smoothness indicator, a composition of first derivatives of local smoothness indicators with their respective interpolation polynomials. The sufficient condition of nonlinear weights is satisfied by using Taylor expansions. We verify the proposed scheme through several benchmark test cases for scalar and system of hyperbolic conservation laws. We observe from the numerical simulations that the designed WENO scheme demonstrates high resolutions, fifth-order accuracy, and robustness at critical points as compared with other modern WENO schemes.

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space and an explicit strong stability preserving Runge-Kutta scheme in time. When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues. The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way. For piecewise constant polynomials in space and explicit Euler in time, the stabilized scheme is monotone for scalar problems. For higher polynomial degrees p, our numerical experiments show convergence orders of p+1 for smooth flow and robust behavior in the presence of shocks.

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

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

Quantity of the inverse problem data for the system of conservation laws

In this paper we study properties of the model, that describes the plane acoustic waves propagation. The model is based on the hyperboliv system of PDE, which is solved numerically by using the finite-volume method, based on Godunov scheme. After studying the direct problem we turn to the inverse one, where our goal is to recover the parameters of the system of PDE by using the initial data, measured in the receivers. We obtain the formula for the gradient of the misfits functional, which allows us to apply gradient-based optimization for recovering the density of the medium. We present the results of numerical experiments for different number of receivers, thus, studying the influence of the quantity of the data of inverse problem on the accuracy of the solution.

On the Systems of Conservation Laws and on a New Way To Construct for them Neural Networks Algorithms

On the Systems of Conservation Laws and on a New Way To Construct for them Neural Networks Algorithms

Selective decay for the rotating shallow-water equations with a structure-preserving discretization

Numerical models of weather and climate critically depend on the long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2), 495 (2013)], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [Pavlov et al., “Structure-preserving discretization of incompressible fluids,” Physica D 240(6), 443–458 (2011)] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out.

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws

Application of the CABARET Scheme for Calculating Discontinuous Solutions of a Hyperbolic System of Conservation Laws

Application of the CABARET Scheme for Calculating Discontinuous Solutions of a Hyperbolic System of Conservation Laws

The multiplication of distributions in the study of delta shock wave for a hyperbolic system of Temple class

The Riemann problem for a hyperbolic system of conservation laws belonging to Temple class is considered within frame of [Formula: see text]-solutions based on the concept of solutions defined in the setting of a distributional product. We obtain all the possible discontinuous solutions within a convenient space of distributions that contains Dirac delta measures and discontinuous functions. In addition, an interesting discontinuous solution is found for certain Riemann initial conditions, in which the Dirac delta measure is involved in both of the two state variables simultaneously. As an application, a detailed example for the mentioned system but involving a linear damping term is provided, in which the trajectories of discontinuous solutions including contact discontinuity, shock wave and delta shock wave change in exponential forms.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

A zero-noise limit to a symmetric system of conservation laws

We consider a symmetric system of conservation laws and its stochastic approximation with stochastic multiplicative noise. Using the vanishing viscosity with the zero-noise limit we obtain a deterministic weak solution for some time interval.

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

Stability of Entropy Solution of Conservation Law Systems on Flow Function and Relaxation Function

Stability of Entropy Solution of Conservation Law Systems on Flow Function and Relaxation Function

Simulations of plasmas and fluids using anti-symmetric models

ALMA (Anti-symmetric, Large-Moment, Accelerated) is a fast, flexible, and scalable toolkit designed to solve hyperbolic conservation law systems in hybrid supercomputers. This manuscript describes the theoretical background and implementation of ALMA, which uses the anti-symmetric formulation of fluids to obtain simple, robust, and easily paralellizable code. Practical GPU acceleration is realized on entire applications with an overall gain factor of 2 to 4. ALMA also provides a parallel, GPU accelerated sparse solver based on geometric multigrid, capable of diagonalizing linear systems with 239 unknowns. We demonstrate ALMA's scaling and performance in petascale supercomputers and use standard fluid models to verify the overall approach with canonical benchmark problems.

Lax–Wendroff consistency of finite volume schemes for systems of non linear conservation laws: extension to staggered schemes

We prove in this paper the Lax–Wendroff consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax–Wendroff theorem for general colocated or non-colocated schemes. This result is obtained for general polygonal or polyhedral meshes, under assumptions which, for usual practical cases, essentially boil down to a flux-consistency constraint; this latter is, up to our knowledge, novel and compares the discrete flux at a face to the mean value over the adjacent cell of the continuous flux function applied to the discrete unknown function. We first briefly show how this result copes with multipoint colocated schemes on general meshes. We then apply it to prove the consistency of a finite volume discretisation of a convection operator featuring a (convected) scalar variable and a (convecting) velocity field, with a staggered approximation, i.e. with a cell-centred approximation of the scalar variable and a face-centred approximation of the velocity.

Oscillation mitigation of hyperbolicity-preserving intrusive uncertainty quantification methods for systems of conservation laws

In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. While intrusive methods inherit certain advantages such as adaptivity and an improved accuracy, they suffer from two key issues. First, intrusive methods tend to show oscillations, especially at shock structures and second, standard intrusive methods can lose hyperbolicity. The aim of this work is to tackle these challenges with the help of two different strategies. First, we combine filters with the multi-element approach for the hyperbolicity-preserving stochastic Galerkin (hSG) scheme. While the limiter used in the hSG scheme ensures hyperbolicity, the filter as well as the multi-element ansatz mitigate oscillations. Second, we derive a multi-element approach for the intrusive polynomial moment (IPM) method. Even though the IPM method is inherently hyperbolic, it suffers from oscillations while requiring the solution of an optimization problem in every spatial cell and every time step. The proposed multi-element IPM method leads to a decoupling of the optimization problem in every multi-element. Thus, we are able to significantly decrease computational costs while improving parallelizability. Both proposed strategies are extended to adaptivity, allowing to adapt the number of basis functions in each multi-element to the smoothness of the solution. We finally evaluate and compare both approaches on various numerical examples such as a NACA airfoil and a nozzle test case for the two-dimensional Euler equations. In our numerical experiments, we observe the mitigation of spurious artifacts. Furthermore, using the multi-element ansatz for IPM significantly reduces computational costs.

The WAF scheme for the isentropic drift-flux model of compressible two-phase flows

This paper focuses on the extension of the Weighted Average Flux (WAF) scheme for the numerical simulation of two-phase gas–liquid flow by imposing velocity equilibrium and without mechanical equilibrium of the transient drift-flux model. The model becomes a hyperbolic system of conservation laws with realistic closure relations where both phases are strongly coupled during their motion. Exploiting this, the drift-flux model discretization, simulation and investigation becomes very fast, simple and robust. The efficiency of the WAF scheme as being a second order in space and time without data reconstruction have been demonstrated in the published literature for compressible single-phase flows. However, the scheme is rarely applied to compressible two-phase flows. Based on a recent and complete exact Riemann solver for the drift-flux model, the model is numerically solved by the WAF scheme. The numerical algorithm accuracy and ability are validated through different published test cases. It is shown that the proposed scheme can be effectively employed to simulate two-phase flows involving discontinuities such as shocks and interfaces. The proposed WAF scheme is also compared with other numerical methods. Simulation results show appropriate agreement of WAF scheme even with the exact solutions. Comparisons of the presented simulations demonstrate that the behaviour of WAF scheme is encouraging, more accurate and fast than other numerical methods.

Positivity preserving and entropy consistent approximate Riemann solvers dedicated to the high-order MOOD-based Finite Volume discretization of Lagrangian and Eulerian gas dynamics

In this paper we propose to revisit the notion of simple Riemann solver both in Lagrangian and Eulerian coordinates following the seminal work of Gallice ”Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates” in Numer. Math., 94, 2003. We provide in this work the relation between the Eulerian and Lagrangian forms of systems of conservation laws in 1D. Then an approximate (simple) Lagrangian Riemann solver for gas dynamics is derived based on the notions of positivity preservation and entropy control. Its Eulerian counterpart is further deduced. Next we build the associated 1D first-order accurate cell-centered Lagrangian Godunov-type Finite Volume scheme and show numerically its behaviors on classical test cases. Then using the Lagrangian–Eulerian relationships, we derive and test the Eulerian Godunov-type Finite Volume scheme, which inherits by construction the properties of the Lagrangian solver in terms of positivity preservation and well-defined CFL condition. At last we extend this Eulerian scheme to arbitrary orders of accuracy using a Runge–Kutta time discretization, polynomial reconstruction and an a posteriori MOOD limiting strategy. Numerical tests are carried out to assess the robustness, accuracy, and essentially non-oscillatory properties of the numerical methods.