One-dimensional Conservation Laws Research Articles

A deep learning operator-based numerical scheme method for solving 1D wave equations

Abstract In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods.

Optimal transport with nonlinear mobilities: A deterministic particle approximation result

Abstract We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ {N\to\infty} to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 {\tau>0} . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.

Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

A High-Resolution Scheme for Solving One-Dimensional Hyperbolic Conservation Law Equations

A High-Resolution Scheme for Solving One-Dimensional Hyperbolic Conservation Law Equations

A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.

Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws

In this paper, we analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov (Ann. Probab. 46 (2018) 1042–1069) to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy (Math. Comp. 73 (2004) 777–812) to check the convergence in L1(R) with rate O(1N+h) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O(1N+h). We provide numerical results which confirm our theoretical estimates.

Modeling of Traffic Flow on Roundabouts

The aim of this research is to create the modeling of roundabouts. First of all, the three-arm roundabout is created for validation with the existing model. Then, the model is expanded to the four-arm roundabout. In the development of a modern intelligent transportation system, the effectiveness of dealing with the non-linear, time-varying and congested traffic flow is imperative in achieving traffic control and accuracy. In this paper, the roundabout is modelled as a circuit of 2×2 junction comprising a main lane and a secondary lane. The rotation of the roundabout is in the clockwise direction, as in the case of Malaysia. In mathematical modelling, the traffic flow is created, based on one-dimensional hyperbolic conservation laws which are represented by non-linear partial differential equations where the unknown variable is a conserved quantity. As a scheme used in the computation and analysis, the Godunov method computes the fluxes at the interfaces of each cell in order to advance the solution of a Riemann Problem. In addition, the Courant-Friedrichs-Levy (CFL) condition is proposed and used to ensure the stability and accuracy of the numerical algorithm where the time step is not a constant. The optimization on the roundabout for Total Travel Time and Total Waiting Time with several parameters is applied to generate numerous results which will assist in assessing the reasonableness of the roundabout. The comparison data of the three-arm roundabout with our model and the existing model are discussed. In comparison, our results show similar properties with higher readings than in other published papers because our calculations involved all arms and roads. In addition, the comparison data between three-arm and four-arm roundabouts are reasonable and logical. Lastly, our model is more flexible and realistic, as compared to the existing model.

Propagating fronts for a viscous Hamer-type system

<p style='text-indent:20px;'>Motivated by radiation hydrodynamics, we analyse a <inline-formula><tex-math id="M1">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named <b>viscous Hamer-type system</b>. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called <i>sub-shock</i>– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on <i>Geometric Singular Perturbation Theory</i> (GSPT) as introduced in the pioneering work of Fenichel [<xref ref-type="bibr" rid="b5">5</xref>] and subsequently developed by Szmolyan [<xref ref-type="bibr" rid="b21">21</xref>]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.</p>

Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-D convex scalar viscous conservation laws

This paper studies the asymptotic stability of shock profiles and rarefaction waves under space-periodic perturbations for one-dimensional convex scalar viscous conservation laws. For the shock profile, we show that the solution approaches the background shock profile with a constant shift in the $ L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting to the case of localized perturbations is that the constant shift cannot be determined by the initial excessive mass in general, which indicates that the periodic oscillations at infinities make contributions to this shift. And the vanishing viscosity limit for the shift is also shown. The key elements of the poof consist of the construction of an ansatz which tends to two periodic solutions as $ x \rightarrow \pm\infty, $ respectively, and the anti-derivative variable argument, and an elaborate use of the maximum principle. For the rarefaction wave, we also show the stability in the $ L^\infty(\mathbb{R}) $ norm.

A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

Numerical solutions in chromatography using large time step and overlapping grids methods

We consider two systems of one-dimensional conservation laws that describe the process of column chromatography in chemistry in isolating a single compound from a mixture. We use recently proposed large time step and overlapping grids finite volume numerical methods in approximating solutions to a variety of initial value problems resulting in classical solutions as well as in singular and $$\delta $$ -shock solutions. The novel idea presented in the large time step method reduces the number of time steps needed to reach the final time and, thus, allows faster marching in the time direction. The overlapping grids methods are crucial when considering problems where an object is covered with multiple grids. It is important to ensure that the numerical method is constructed in such a way that it is conservative on the overlap and, moreover, that the approximate solutions converge to the weak solution, and in the case of a scalar equation, to the entropy solution. The main contribution of this paper is to show effectiveness of the proposed numerical methods for approximating solutions to systems of equations since their convergence was rigorously proved so far only in case of a scalar equation.

Viscous conservation laws in 1D with measure initial data

The one-dimensional viscous conservation law is considered on the whole line u t + f ( u ) x = ε u x x , ( x , t ) ∈ R × R + ¯ , ε > 0 , \begin{equation*} u_t + f(u)_x=\varepsilon u_{xx},\quad (x,t)\in \mathbb {R}\times \overline {\mathbb {R}_{+}},\quad \varepsilon >0, \end{equation*} subject to positive measure initial data. The flux f ∈ C 1 ( R ) f\in C^1(\mathbb {R}) is assumed to satisfy a p − p- condition, a weak form of convexity. In particular, any flux of the form f ( u ) = ∑ i = 1 J a i u m i f(u)=\sum _{i=1}^Ja_iu^{m_i} is admissible if a i > 0 , m i > 1 , i = 1 , 2 , … , J . a_i>0,\,m_i>1,\,\,i=1,2,\ldots ,J. The only case treated hitherto in the literature is f ( u ) = u m f(u)=u^m [Arch. Rat. Mech. Anal. 124 (1993), pp. 43–65] and the initial data is a “single source”, namely, a multiple of the delta function. The corresponding solutions have been labeled as “source-type” and the treatment made substantial use of the special form of both the flux and the initial data. In this paper existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for the viscous Hamilton-Jacobi equation. Some estimates are independent of the viscosity coefficient, thus leading to new estimates for the (inviscid) hyperbolic conservation law.

一维双曲守恒律方程的保极值间断有限体积元方法

一维双曲守恒律方程的保极值间断有限体积元方法

Shape-Based Nonlinear Model Reduction for 1D Conservation Laws

We present a novel method for model reduction of one-dimensional conservation law to the dynamics of the parameters describing the approximate shape of the solution. Depending on the parametrization, each parameter has a well-defined physical meaning. The obtained ODE system can be used for the estimation and control purposes. The model reduction is performed by minimizing the divergence of flows between the original and reduced systems, and we show that this is equivalent to the minimization of the Wasserstein distance derivative. The method is then tested on the heat equation and on the LWR (Lighthill-Whitham-Richards) model for vehicle traffic.

Detecting troubled-cells on two-dimensional unstructured grids using a neural network

In a recent paper (Ray and Hesthaven, 2018) [38], we proposed a new type of troubled-cell indicator to detect discontinuities in the numerical solutions of one-dimensional conservation laws. This was achieved by suitably training an artificial neural network on canonical local solution structures for conservation laws. The proposed indicator was independent of problem-dependent parameters, giving it an advantage over existing limiter-based indicators. In the present paper, we extend this approach to train a similar network capable of detecting troubled-cells on two-dimensional unstructured grids. The proposed network has a smaller architecture compared to its one-dimensional predecessor, making it computationally efficient. Several numerical results are presented to demonstrate the performance of the new indicator.

A Fourth Order Entropy Stable Scheme for Hyperbolic Conservation Laws

This paper develops a fourth order entropy stable scheme to approximate the entropy solution of one-dimensional hyperbolic conservation laws. The scheme is constructed by employing a high order entropy conservative flux of order four in conjunction with a suitable numerical diffusion operator that based on a fourth order non-oscillatory reconstruction which satisfies the sign property. The constructed scheme possesses two features: (1) it achieves fourth order accuracy in the smooth area while keeping high resolution with sharp discontinuity transitions in the nonsmooth area; (2) it is entropy stable. Some typical numerical experiments are performed to illustrate the capability of the new entropy stable scheme.

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.

A moving mesh WENO method based on exponential polynomials for one-dimensional conservation laws

In the article [Yang et al. (2012) [37]], the authors have developed a high order moving mesh WENO method for one-dimensional (1D) hyperbolic conservation laws, which is shown to be effective in resolving shocks and other complex solution structures. In this paper, in the light of the similar moving mesh technique, we develop a novel WENO scheme with non-polynomial bases, in particular, the exponential bases to further improve the performance of WENO schemes for solving 1D conservation laws. Furthermore, we modify the original moving mesh technique by developing a new monitor function as well as a different mesh smoothing strategy. A collection of numerical examples is presented to demonstrate high order accuracy and robustness of the method in capturing smooth and non-smooth solutions including the strong δ shock arising from the weakly hyperbolic pressureless Euler equations.

Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux

Solutions to a class of one-dimensional conservation laws with discontinuous flux are constructed relying on the Crandall--Liggett theory of nonlinear contractive semigroups [H. Brezis and A. Pazy,...

A conservation law with multiply discontinuous flux modelling a flotation column

Flotation is a unit operation extensively used in the recovery of valuable minerals in mineral processing and related applications. Essential insight to the hydrodynamics of a flotation column can be obtained by studying just two phases: gas and fluid. To this end, the approach based on the drift-flux theory, proposed in similar form by several authors, is reformulated as a one-dimensional non-linear conservation law with a multiply discontinuous flux. The unknown is the gas volume fraction as a function of height and time, and the flux function depends discontinuously on spatial position due to several feed inlets. The resulting model is similar, but not equivalent, to previously studied clarifier-thickener models for solid-liquid separation and therefore adds a new real-world application to the field of conservation laws with discontinuous flux. Steady-state solutions are studied in detail, including their construction by applying an appropriate entropy condition across each flux discontinuity. This analysis leads to operating charts and tables collecting all possible steady states along with some necessary conditions for their feasibility in each case. Numerical experiments show that the transient model recovers the steady states, depending on the feed rates of the different inlets.