Two-dimensional Conservation Laws Research Articles

A class of bound-preserving MUSCL-Hancock schemes in two dimensions

In this paper we discuss the MUSCL-Hancock scheme for the two-dimensional conservation laws and derive the bound-preserving conditions by a parameterized convex decomposition method which is proposed in [Journal of Computational Physics, 474:111805, 2023] for one-dimensional problems. The CFL number equals the half of the one-dimensional value and then ensures the speed-up compared with the classical two-stage MUSCL scheme. The bound-preserving slope limiter on the preliminary reconstruction becomes problem dependent. The maximum-minimum preserving slope limiter for the scalar problems becomes the same with its one-dimensional version. A positivity-preserving slope limiter for the Euler system is designed including a different parameter from the scalar problems and more efficient operators than the existing limiters. Numerical experiments are reported to demonstrate the robustness of the proposed scheme.

Registration-based model reduction of parameterized two-dimensional conservation laws

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping Φ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

Controlling oscillations in spectral methods by local artificial viscosity governed by neural networks

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods since a local estimate of the solution regularity is generally required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this, in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations.

A new WENO based Chebyshev Spectral Volume method for solving one- and two-dimensional conservation laws

High-order methods have attracted considerable attention in the CFD fields because of their potential in achieving higher accuracy with a lower cost than low order methods. In this paper, a new class of high order spectral volume method based on Chebyshev polynomials as an approximation function called Chebyshev Spectral Volume (CSV) method is presented for hyperbolic conservation laws in quadrilateral meshes. As we know, the hyperbolic conservation law may contain discontinuities even if the initial conditions are smooth. To capture these discontinuities, we propose a new limiter to reconstruct the numerical approximation on the target cell, which is marked as a troubled cell. It is based on the Chebyshev polynomials in the target and its immediate neighboring cells to achieve a high order of accuracy and non-oscillatory properties. Also, a new indicator is introduced to detect troubled cells. This indicator depends neither on the numerical order of accuracy nor on a problem-dependent parameter. A series of 1D and 2D scalar and system conservation laws are presented to evaluate the performance of the CSV method coupled with the new limiter and indicator.

Two-dimensional macroscopic model for large scale traffic networks

In this article we introduce a new traffic flow model for a dense urban area. We consider a two-dimensional conservation law in which the velocity magnitude is given by the fundamental diagram and the velocity direction is constructed following the network geometry and assuming we do not have precise information of drivers trajectories. We validate the model using synthetic data from Aimsun and propose a reconstruction technique to recover the 2D density from the data of individual vehicles. A comparison between the model and the data is shown.

Application of Green−Naghdi Equations for Modeling Wave Flows with Undular Bores

The basic conservation laws in the Green–Naghdi model of shallow-water theory are derived from the two-dimensional integral conservation laws of mass and the total momentum describing the plane-parallel flow in an ideal incompressible fluid above a horizontal bottom. This conclusion is based on the concept of a local hydrostatic approximation, which generalizes the concept of the long-wavelength approximation and is used for analyzing the applicability of the Green–Naghdi equations in modeling the wave flows of a fluid with undular bores.

Maximum-Principle-Satisfying and Positivity-Preserving High Order Central DG Methods on Unstructured Overlapping Meshes for Two-Dimensional Hyperbolic Conservation Laws

In this paper, we first present a family of high order central discontinuous Galerkin methods defined on unstructured overlapping meshes for the two-dimensional conservation laws. The primal mesh is a triangulation of the computational domain, while the dual mesh is a quadrangular partition which is formed by connecting an interior point and the three vertexes of each triangle on the primal mesh. We prove the \(L^2\) stability of the present method for linear equation. Then we design and analyze high order maximum-principle-satisfying central discontinuous Galerkin methods for two-dimensional scalar conservation law, and high order positivity-preserving central discontinuous Galerkin methods for two-dimensional compressible Euler systems. The performance of the proposed methods is finally demonstrated through a set of numerical experiments.

Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics

The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $\epsilon$ embedded in $\mathbb E^3$. In such a system, two-dimensional conservation law needs an additional term just like an anomaly. The additional term can be obtained by the $\epsilon$ expansion. This term has a Gaussian and a mean curvature dependence and can be written as the total divergence of geometric flow $J^i_{G}$. This fact holds in both classical and quantum mechanics.

Envelope and classification of global structures of solutions for a class of two-dimensional conservation laws

Two-dimensional Riemann problem for scalar conservation law are investigated and classification of global structure for its non-selfsimilar solution is given by analysis of structure and classification of envelope for non-selfsimilar 2D rarefaction wave. Initial data has two different constant states which are separated by initial discontinuity. We propose the concepts of plus envelope, minus envelope and mixed envelope, and some new structures and evolution phenomena are discovered by use of these concepts.

A family of finite volume Eulerian–Lagrangian methods for two-dimensional conservation laws

We develop a family of finite volume Eulerian–Lagrangian methods for the solution of nonlinear conservation laws in two space dimensions. The proposed approach belongs to the class of fractional-step procedures where the numerical fluxes are reconstructed using the modified method of characteristics, while an Eulerian method is used to discretize the conservation equation in a finite volume framework. The method is simple, accurate, conservative and it combines advantages of the modified method of characteristics to accurately solve the nonlinear conservation laws with an Eulerian finite volume method to discretize the equations. The proposed finite volume Eulerian–Lagrangian methods are conservative, non-oscillatory and suitable for hyperbolic or non-hyperbolic systems for which Riemann problems are difficult to solve or do not exist. Numerical results are presented for an advection–diffusion equation with known analytical solution. The performance of the methods is also analyzed on several applications in Burgers and Buckley–Leverett problems. The aim of such a method compared to the conventional finite volume methods is to solve nonlinear conservation laws efficiently and with an appropriate level of accuracy.

CWENO-type entropy consistent scheme for two dimensional scalar hyperbolic conservation laws

This paper advanced the Central Weighted Essentially Nonoscillatory(CWENO)-type entropy consistent schemes to simulate the two-dimensional conservation laws of the initial boundary value problem.The numerical results were analyzed and compared with the exact solutions.It is pointed out that the courant-friedrich-lewy can attain 0.6 and shock transition zone is one or two cells.The results indicate that the new numerical method in this paper has high-resolution and strong stability.

Cauchy problem for multiscale conservation laws: Application to structured cell populations

In this paper, we study a vector conservation law that models the growth and selection of ovarian follicles. This work is motivated by a multiscale mathematical model. A two-dimensional conservation law describes the age and maturity structuration of the follicular cell populations. The densities interact through a coupled hyperbolic system between different follicles and cell phases, which results in a vector conservation law and coupling between boundary conditions. The maturity velocity functions possess both a local and a nonlocal character. We prove the existence and uniqueness of the weak solution to Cauchy problem with bounded initial and boundary data.

Numerical simulation of inviscid bubble dynamics in a centrally symmetric gravitational field

The motion of bubbles in a centrally symmetric gravitational field is numerically simulated using two-dimensional conservation laws (Euler equations). The dynamics of bubbles with various numbers of modes in the initial perturbation are studied. The numerical results reveal features that are substantially different from the plane case in a homogeneous gravitational field. Bubble perturbations nearly do not interact at the formation stage. The lowest modes are amplified in the course of the bubble evolution.

Alternating Evolution Schemes for Hyperbolic Conservation Laws

The alternating evolution (AE) system of Liu [J. Hyperbolic Differ. Equ., 5 (2008), pp. 421-447], $\partial_t u+ \partial_x f(v) =\frac{1}{\epsilon}(v-u)$, $\partial_t v + \partial_x f(u) =\frac{1}{\epsilon}(u-v)$, serves as a refined description of systems of hyperbolic conservation laws $\partial_t \phi+ \partial_x f(\phi)=0$, $\phi(x, 0)=\phi_0(x)$. The solution of conservation laws is precisely captured when two components take the same initial value as $\phi_0$, or is approached by two components exponentially fast when $\epsilon \downarrow 0$ if two initial states are sufficiently close. This nice property enables us to construct novel shock capturing schemes by sampling the AE system on alternating grids. In this paper we develop a class of local AE schemes by taking advantage of the AE system. Our approach is based on an average of the AE system over a hypercube centered at $x$ with vertices at $x\pm\Delta x$. The numerical scheme is then constructed by sampling the averaged system over alternating grids. Higher order accuracy is achieved by a combination of high order nonoscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scale parameter $\epsilon$ reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. For the first and second order local AE schemes applied to one-dimensional scalar conservation laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing property. The formulation procedure of AE schemes in multiple dimensions is given, followed by both the first and second order AE schemes for two-dimensional conservation laws. Numerical experiments for both scalar conservation laws and compressible Euler equations are presented to demonstrate the high order accuracy and capacity of these AE schemes.

New structures for non-selfsimilar solutions of multi-dimensional conservation laws

In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of two-dimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u + > 0, while explicit non-selfsimilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u + < 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.

Dynamic continuum pedestrian flow model with memory effect

In this paper, we develop a macroscopic model for pedestrian flow using the dynamic continuum modeling approach. We consider a two-dimensional walking facility that is represented as a continuum within which pedestrians can move freely in any direction. A pedestrian chooses a route based on his or her memory of the shortest path to the desired destination when the facility is empty and, at the same time, tries to avoid high densities. In this model, pedestrian flow is governed by a two-dimensional conservation law, and a general speed-flow-density relationship is considered. The model equation is solved numerically using the discontinuous Galerkin method, and a numerical example is employed to demonstrate both the model and the effectiveness of the numerical method.

Numerical Simulation of Highly Underexpanded Axisymmetric Jet with Runge-Kutta Discontinuous Galerkin Finite Element Method

Highly underexpanded axisymmetric jet was simulated using the Runge-Kutta Discontinuous Galerkin (RKDG) finite element method, which, based on two-dimensional conservation laws, was used to solve the axisymmetric Euler equations. The computed results show that the complicated flow field structures of interest, including shock waves, slipstreams and the triple point observed in experiments could be well captured using the RKDG finite element method. Moreover, comparisons of the Mach disk location exhibit excellent agreements between the computed results and experimental measurements, indicating that this method has high capability of capturing shocks without numerical oscillation and artificial viscosity occurring near the discontinuous point.

A Cartoon for the Climbing Ramp Problem of a Shock and von Neumann Paradox

The climbing ramp problem of a shock for two-dimensional scalar conservation laws is considered. By use of the generalized characteristic analysis method, we prove constructively that there are only three configurations of solutions: regular reflection-like, Mach reflection-like and von Neumann reflection-like for a kind of flux function of the conservation laws and some initial data. The criteria for these three configurations are obtained. Furthermore, a necessary condition for the appearance of regular reflection in gas dynamics is presented.

COMPENSATED COMPACTNESS FOR 2D CONSERVATION LAWS

We introduce a new framework for studying two-dimensional conservation laws by compensated compactness arguments. Our main result deals with 2D conservation laws which are nonlinear in the sense that their velocity fields are a.e. not co-linear. We prove that if uε is a family of uniformly bounded approximate solutions of such equations with H-1-compact entropy production and with (a minimal amount of) uniform time regularity, then (a subsequence of) uε convergences strongly to a weak solution. We note that no translation invariance in space — and in particular, no spatial regularity of u(·, t) is required. Our new approach avoids the use of a large family of entropies; by a judicious choice of entropies, we show that only two entropy production bounds will suffice. We demonstrate these convergence results in the context of vanishing viscosity, kinetic BGK and finite volume approximations. Finally, the intimate connection between our 2D compensated compactness arguments and the notion of multi-dimensional nonlinearity based on kinetic formulation is clarified.

SELF-SIMILAR SOLUTIONS OF TWO-DIMENSIONAL CONSERVATION LAWS

Self-similar reduction of an important class of two-dimensional conservation laws leads to boundary value problems for equations which change type. We have established a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution. This paper summarizes the principal features of the method.A preliminary version of these notes formed the basis of a series of three lectures at the Newton Institute in April, 2003. They are a report of research carried out jointly with Sunčica Čanić, Eun Heui Kim and Gray Lieberman.