One-dimensional Hyperbolic 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.

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.

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.

A Semi-Lagrangian Godunov-Type Method without Numerical Viscosity for Shocks

One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is a numerical viscosity for shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The proposed method belongs to a group of particle-in-cell (PIC) methods; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. This article describes the application of the aforementioned method for the inviscid Burgers’ equation, adiabatic gas dynamics equations, and the one-dimensional scalar hyperbolic conservation law. The numerical analysis results for several test cases (e.g., the standard shock-tube problem of Sod, the Riemann problem of Lax, the double expansion wave problem, the Shu–Osher shock-tube problem) are compared with the exact solution and Harten’s data. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10−5 to find the pressure difference between the current and previous iterations.

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

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.

Axial magnetic field and toroidally streaming fast ions in the dense plasma focus are natural consequences of conservation laws in the curved axisymmetric geometry of the current sheath. II. Towards a first principles theory

This paper continues earlier discussion [S. K. H. Auluck, Phys. Plasmas 21, 102515 (2014)] concerning the formulation of conservation laws of mass, momentum, and energy in a local curvilinear coordinate system in the dense plasma focus. This formulation makes use of the revised Gratton-Vargas snowplow model [S. K. H. Auluck, Phys. Plasmas 20, 112501 (2013)], which provides an analytically defined imaginary surface in three dimensions which resembles the experimentally determined shape of the plasma. Unit vectors along the local tangent to this surface, along the azimuth, and along the local normal define a right-handed orthogonal local curvilinear coordinate system. The simplifying assumption that physical quantities have significant variation only along the normal enables writing laws of conservation of mass, momentum, and energy in the form of effectively one-dimensional hyperbolic conservation law equations using expressions for various differential operators derived for this coordinate system. This formulation demonstrates the highly non-trivial result that the axial magnetic field and toroidally streaming fast ions, experimentally observed by multiple prestigious laboratories, are natural consequences of conservation of mass, momentum, and energy in the curved geometry of the dense plasma focus current sheath. The present paper continues the discussion in the context of a 3-region shock structure similar to the one experimentally observed: an unperturbed region followed by a hydrodynamic shock containing some current followed by a magnetic piston. Rankine-Hugoniot conditions are derived, and expressions are obtained for the specific volumes and pressures using the mass-flux between the hydrodynamic shock and the magnetic piston and current fraction in the hydrodynamic shock as unknown parameters. For the special case of a magnetic piston that remains continuously in contact with the fluid being pushed, the theory gives closed form algebraic results for the fraction of current flowing in the hydrodynamic shock, specific volume, pressure, and fluid velocity of the hydrodynamic shock region, the tangential, normal, and azimuthal components of velocity in the magnetized plasma, the density of the magnetized plasma, the normal and tangential components of the magnetic field, and the tangential, normal, and azimuthal components of the electric field. This explains the occurrence of azimuthally streaming high energy deuterons experimentally observed by Frascati and Stuttgart. The expression derived for the azimuthal component of vector potential can serve as the basis for a proposed experimental test of the theory.

Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Numerical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.

Stability analysis of the inverse Lax–Wendroff boundary treatment for high order upwind-biased finite difference schemes

In this paper, we consider linear stability issues for one-dimensional hyperbolic conservation laws using a class of conservative high order upwind-biased finite difference schemes, which is a prototype for the weighted essentially non-oscillatory (WENO) schemes, for initial–boundary value problems (IBVP). The inflow boundary is treated by the so-called inverse Lax–Wendroff (ILW) or simplified inverse Lax–Wendroff (SILW) procedure, and the outflow boundary is treated by the classical high order extrapolation. A third order total variation diminishing (TVD) Runge–Kutta time discretization is used in the fully discrete case. Both GKS (Gustafsson, Kreiss and Sundström) and eigenvalue analyses are performed for both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate the stability results predicted by the analysis.

Axial magnetic field and toroidally streaming fast ions in the dense plasma focus are natural consequences of conservation laws in the curved axisymmetric geometry of the current sheath

Direct measurement of axial magnetic field in the PF-1000 dense plasma focus (DPF), and its reported correlation with neutron emission, call for a fresh look at previous reports of existence of axial magnetic field component in the DPF from other laboratories, and associated data suggesting toroidal directionality of fast ions participating in fusion reactions, with a view to understand the underlying physics. In this context, recent work dealing with application of the hyperbolic conservation law formalism to the DPF is extended in this paper to a curvilinear coordinate system, which reflects the shape of the DPF current sheath. Locally unidirectional shock propagation in this coordinate system enables construction of a system of 7 one-dimensional hyperbolic conservation law equations with geometric source terms, taking into account all the components of magnetic field and flow velocity. Rankine-Hugoniot jump conditions for this system lead to expressions for the axial magnetic field and three components of fluid velocity having high ion kinetic energy.

Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(⋅), we are able to model the capacity drop phenomenon at the exit. Existence and stability results for the Cauchy problem with Lipschitz constraint function p(⋅) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(⋅). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(⋅) may induce non-uniqueness and instability of solutions.

An approximate linearized characteristic Riemann solver based on blending of Riemann invariants

A new linearized characteristic upwind approximate Riemann solver is proposed based on the blending of the Riemann invariants. The scheme is based on the linearization of the differential relations along the characteristic curves of the one-dimensional hyperbolic conservation laws. It involves construction of the parameters at the cell interface such that they are consistent with the system of characteristic relations. A third order MUSCL reconstruction was used in the present work. Upwinding in the scheme is carried out by upwinding of the Riemann invariants. In this paper the development of the scheme from the compatibility relations for the Euler equations is described. The scheme was then applied to solve various cases of shock tube problems, two-dimensional Riemann problems and finally, as a practical example, it was used to carry out LES of a subsonic jet. The scheme was found to perform remarkably well for all the cases and is extremely computationally efficient, thus making it ideal for practical applications where high accuracy and high speed computation are required.

Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case

The concept of multiresolution-based adaptive DG schemes for non-linear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in (Math Comp, doi: 10.1090/S0025-5718-2013-02732-9 , 2013). The key idea is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and non-linear scalar conservation laws.

Riemann problems with non--local point constraints and capacity drop.

In the present note we discuss in details the Riemann problem for a one-dimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the well--posedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non--local operator depending on the solution itself. We provide several explicit examples. We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.

A Moving Mesh WENO Method for One-Dimensional Conservation Laws

In this paper, we develop an efficient moving mesh weighted essentially nonoscillatory (WENO) method for one-dimensional hyperbolic conservation laws. The method is based on the quasi-Lagrange approach of the moving mesh strategy in which the mesh is considered to move continuously in time. Several issues arising from the implementation of the scheme, including mesh smoothness, mesh movement restriction, and computation of transformation relations, and their effects on the accuracy of the underlying scheme have been addressed. Particularly, it is found that a least squares smoothing can be used to effectively smooth the mesh, and the transformation relations can be computed using either high order finite differences or WENO applied to some geometric conservation laws. Moreover, mesh movement can cause WENO schemes to become unconditionally unstable. A simple strategy is used to restrict the mesh movement and recover the stability. Numerical results are presented to demonstrate the accuracy and shock-captu...

Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems

In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error.

A Finite Element, Multiresolution Viscosity Method for Hyperbolic Conservation Laws

It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flow. It is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant approach, referred to as spectral viscosity methods, has been developed for spectral methods in which one adds diffusion only to the high-frequency modes of the solution, the result being that stabilization is effected without sacrificing accuracy. We extend this idea into the finite element framework by using hierarchical finite element functions as a multifrequency basis. The result is a new finite element method for solving hyperbolic conservation laws in which artificial diffusion can be applied selectively only to the high-frequency modes of the approximation. As for spectral viscosity methods, this results in stability without compromising accuracy. In the context of a one-dimensional scalar hyperbolic conservation law, we prove the convergence of approximate solutions, obtained using the new method, to the entropy solution of the conservation law. To illustrate the method, the results of a computational experiment for a one-dimensional hyperbolic conservation law are provided.