Nonconvex Flux Function Research Articles

A novel approach to solve hyperbolic Buckley-Leverett equation by using a transformer based physics informed neural network

Solving hyperbolic partial differential equation is a challenging task due to the non-linear feature that requires to capture the shock wave. Numerical solution relies on discretization of both spatial and temporal domain, and iterative approach like Newton's method is involved and time step size is crucial for stability and convergence in the presence of non-linearity. Physics-informed neural networks (PINNs) offer a new and versatile approach for solving partial different equations by minimizing the residual of governing equations and approaching to the initial and boundary conditions. Currently, most PINNs are built based on a simple fully connected neural network which exhibits some limitations to model complex non-linear partial differential equations. In this paper, a novel method is developed to combine Transformer model and PINNs approach (Tr-PINN) to solving a hyperbolic partial differential equation directly without any prior knowledge. Tr-PINN method is based on a series of Transformer blocks where self-attention mechanism is used to capture the non-linearity features of the solution. Unlike most PINNs models generate inputs with spatial and temporal vectors only, Tr-PINN introduces the non-linearity term mobility ratio as additional input vector. The method is tested on a classical hyperbolic problem, called Buckley-Leverett equation with non-convex flux function. We found that the Tr-PINN method can capture the water shock front effectively and provide a general solution for Buckley-Leverett equation under various mobility ratio conditions.

Subcell limiting strategies for discontinuous Galerkin spectral element methods

We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre–Gauss–Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods.By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach.We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the {restricted} case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial-boundary-value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.

A minimization approach to conservation laws with random initial conditions and non-smooth, non-strictly convex flux

We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) =\left\vert p\right\vert ^{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x,t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process by taking the continuum limit as the partition becomes more fine.

Entropy Solutions of a Scalar Conservation Law Modeling Sedimentation in Vessels With Varying Cross-Sectional Area

The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a nonconvex flux function and a weight function that depends on spatial position. The sought unknown is the local solids' volume fraction. For the most important cases of vessels with downward-decreasing cross-sectional area and flux functions with at most one inflection point, entropy solutions of this problem are constructed by the method of characteristics. Solutions exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the space-time plane. These trajectories are given by ordinary differential equations that arise from the jump condition. It is shown that three qualitatively different solutions may occur in dependence of the initial concentration. The potential application of the findings is a new method of flux identification via settling tests in a suitably shaped vessel. ...

The Computation of Nonclassical Shock Waves in Porous Media with a Heterogeneous Multiscale Method: The Multidimensional Case

We consider two-phase flow problems in porous media with overshooting waves. If higher-order effects are neglected, such problems are governed by a hyperbolic conservation law with a nonconvex flux function as the macroscale model. It is well known that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or in a second approach an extended system, taking rate-dependent capillary pressure effects into account. We present a new multidimensional mass-conserving numerical method to solve the macroscale model. The method belongs to the class of heterogeneous multiscale methods in the sense of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] and is $L^{\infty}$-stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. [Interfaces Free Bound., 10 (2008), pp. 399--421]. Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.

IMEX WENO Schemes for Two-phase Flow Vertical Equilibrium Processes in a Homogeneous Porous Medium

The motion of two-phase flow in a porous medium under the condition of ver tical equilibrium can be described by a viscous conservation law that involves a non-convex flux function with two inflection points. In (5), a first order Godunov scheme was used to numerically approximate solutions of the model. In this paper we show that using instead the high resolution Weighted Essentially Non Oscillatory (WENO) technology, and an IMEX strategy to handle the capillary term by an implicit discretization, leads to a noticeable increase in resolution power and efficiency. We carefully discuss the imp lementation of WENO schemes for the model equation, paying special attention to the choice of the definition of the numerical viscos ity. We also present numerical simulations when the capillary number is negligible (i.e., the model is a homogeneous conservation law) and non-negligible (i.e. the model equation becomes a 'viscous' conservation law). The numerical results are compared w ith those obtained with the method proposed in (5) in terms of accuracy, resolution power and global efficiency.

Classical solutions to the scalar conservation law with discontinuous initial data

Sufficient and necessary conditions for the existence and uniqueness of classical solutions to the Cauchy problem for the scalar conservation law are found in the class of discontinuous initial data and non-convex flux function. Regularity of rarefaction

Application of the Chebyshev pseudospectral method to van der Waals fluids

In this paper, we consider a class of van der Waals flows with non-convex flux functions. In these flows, nonclassical under-compressive shock waves can develop. Such waves, which are characterized by kinetic functions, violate classical entropy conditions. We propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to 10th-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes.

Mini-Workshop: Hyperbolic Aspects of Phase Transition Dynamics

The workshop “Hyperbolic aspects of phase-transition dynamics”, organized by Rinaldo M. Colombo (Brescia), Dietmar Kröner (Freiburg) and Philippe G. LeFloch (Paris) was held February 24th–March 1st, 2008. We had 15 participants from five different countries. The atmosphere in the Oberwolfach Research Institute was very stimulating and has initiated many fruitful discussions and exchange of ideas. The participants of the mini-workshop thank the administration of the institute for the possibility for organize this meeting. The Navier–Stokes equations for van der Waals fluids with viscosity and capillarity effects included form a challenging model of partial differential equations describing important physical phenomena. The associated set of first-order conservation laws is of hyperbolic or hyperbolic-elliptic type, and admits propagating discontinuities (shock waves). Understanding the singular limit when the viscosity and capillarity coefficients vanish is particularly challenging and is required to single out physically admissible, discontinuous solutions to the first-order conservation laws. These solutions turn out to contain both compressive shocks and undercompressive shocks which violate standard entropy criteria and must be characterized via the so-called kinetic relations. There were two talks (N. Bedjaoui, P.G. LeFloch) on conservation laws with non-convex flux functions which are supplemented by nonlinear, possibly singular, diffusive and dispersive terms. Such systems of equations arise in the dynamics of complex materials undergoing phase transitions and in the dynamics of van der Waals fluids, viscosity and capillarity effects included. Questions which are related to existence, uniqueness and the properties of the viscosity or dispersion limits were discussed. In particular the kinetic relations associated with phase dynamics and nonclassical shocks have been considered. Closely related was a contribution of W. Dreyer about the derivation of the kinetic relations from global entropy inequalities. Three other talks (D. Diehl, D. Kröner, C. Rohde) were dealing with the Navier–Stokes–Korteweg model including non-local formulations, the numerical approximation and the special structure of the interface conditions for the pressure fields respectively. In two talks R.M. Colombo and P. Goatin reported on recent results about analytical and numerical techniques for hyperbolic phase transitions/non-classical shocks in models for vehicular or pedestrian traffic. In this model for high speeds (low densities) the flow is free and described by scalar conservation law and for low speeds (high densities) the flow is congested and is described by a 2 \times 2 system. A different model for the phase transition in fluids is used by S. Müller. In order to take into account two different fluids he introduced a new function, the fraction of gas. This is controlled by an additional transport equation. Mass transfer across the interface is neglected. Other models have been considered in the talk of F. Coquel. He has examined the links between a model consisting of two distinct partial differential momentum equations and a single momentum equation for the mixture plus an algebraic closure for governing the relative velocity. H. Fan considered in his contribution a nonlinear system with viscosity terms for the specific volume, the velocity of the fluid and the weight portion of vapour in the liquid/vapour mixture. The simplified version of this model with zero viscosity has been considered in the talk of A. Corli. A global existence result for large data has been obtained. S. Benzoni-Gavage gave a report on the stability analysis for planar subsonic phase boundaries and the existence of linear surface waves. M. Shearer considered the surfactants of a free surface which is transported by the local speed of the free surface but also by diffusion of surfactant molecules on the surface.

New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws

We discuss an extension of the Jiang–Tadmor and Kurganov–Tadmor fully-discrete non-oscillatory central schemes for hyperbolic systems of conservation laws to unstructured triangular meshes. In doing so, we propose a new, “genuinely multidimensional,” non-oscillatory reconstruction—the minimum-angle plane reconstruction (MAPR). The MAPR is based on the selection of an interpolation stencil yielding a linear reconstruction with minimal angle with respect to the horizontal. This means that the MAPR does not bias the solution by using a coordinate direction-by-direction approach to the reconstruction, which is highly desirable when unstructured meshes consisting of elements with (almost) arbitrary geometry are used. To show the “black-box solver” capabilities of the proposed schemes, numerical results are presented for a number of hyperbolic systems of conservation laws (in two spatial dimensions) with convex and non-convex flux functions. In particular, it is shown that, even though the MAPR is neither designed with the goal of obtaining a scheme that satisfies a maximum principle in mind nor is total-variation diminishing (TVD), it provides a robust non-oscillatory reconstruction that captures composite waves accurately.

A spacetime discontinuous Galerkin method for scalar conservation laws

We present a spacetime discontinuous Galerkin (SDG) finite element method for scalar hyperbolic conservation laws. The method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical Godunov flux on each spacetime element. The discrete basis functions are piecewise continuous in spacetime, admitting discontinuities across element boundaries. The resulting Bubnov–Galerkin method is high-order stable for both convex and non-convex flux functions; it does not require stabilization beyond the basic Galerkin projection. The SDG method is applicable to both layered and unstructured spacetime grids. An element-by-element solution scheme delivers O( N) computational complexity (where N is the number of elements) when applied to spacetime meshes that conform to a special causality constraint. We employ two different limiters to control the local overshoot and undershoot that the basic SDG projection generates near shocks.

Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics

The equations of ideal magnetohydrodynamics (MHD) form a non-strict hyperbolic system with a non-convex flux function and admit non-regular, so-called intermediate shocks. The presence of non-regular waves in the MHD system causes the Riemann problem to be not unique in some cases. This paper investigates the uniqueness of Riemann solutions of ideal MHD. To determine uniqueness conditions we discuss the correspondence of non-regular solutions and non-uniqueness. Additionally the structure of the Hugoniot curves and its non-regular behaviour are demonstrated. It follows that the degree of freedom for solving a Riemann problem is reduced in the case of a non-regular solution. From this, we can deduce uniqueness conditions depending on the initial conditions of an MHD Riemann problem. The results also allow one to construct non-unique solutions. We give an example for the case of non-planar initial conditions.

Fractional Rate of Convergence for Viscous Approximation to Nonconvex Conservation Laws

This paper considers the viscous approximations to conservation laws with nonconvex flux function. It is shown that if the entropy solutions are piecewise smooth, then the rate of L 1 - convergence is a fractional number in (0.5, 1). This is in contrast to the corresponding result for the convex conservation laws. Numerical experiments indicate that the theoretical prediction for the convergence rate is optimal.

Diffusive–Dispersive Traveling Waves and Kinetic Relations: Part I: Nonconvex Hyperbolic Conservation Laws

Motivated by the theory of phase transition dynamics, we consider one-dimensional, nonlinear hyperbolic conservation laws with nonconvex flux-function containing vanishing nonlinear diffusive–dispersive terms. Searching for traveling wave solutions, we establish general results of existence, uniqueness, monotonicity, and asymptotic behavior. In particular, we investigate the properties of the traveling waves in the limits of dominant diffusion, dominant dispersion, and asymptotically small or large shock strength. As the diffusion and dispersion parameters tend to 0, the traveling waves converge to shock wave solutions of the conservation law, which either satisfy the classical Oleinik entropy criterion or are nonclassical undercompressive shocks violating it.

The semigroup generated by a Temple class system with non-convex flux function

We consider the Cauchy problem for a nonlinear $n \times n$ system of conservation laws of Temple class, i.e., with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.

A characteristic based numerical method with tracking for nonlinear wave equations

Many physical problems can be modeled by scalar, first-order, nonlinear, hyperbolic, partial differential equations (PDEs). The solutions to these PDEs often contain shock and rarefaction waves, where the solution becomes discontinuous or has a discontinuous derivative. One can encounter difficulties using traditional finite difference methods to solve these equations. In this paper, we introduce a numerical method for solving first-order scalar wave equations. The method involves solving ordinary differential equations (ODEs) to advance the solution along the characteristics and to propagate the characteristics in time. Shocks are created when characteristics cross, and the shocks are then propagated by applying analytical jump conditions. New characteristics are inserted in spreading rarefaction fans. New characteristics are also inserted when values on adjacent characteristics lie on opposite sides of an inflection point of a nonconvex flux function. Solutions along characteristics are propagated using a standard fourth-order Runge-Kutta ODE solver Shocks waves are kept perfectly sharp. In addition, shock locations and velocities are determined without analyzing smeared profiles or taking numerical derivatives. In order to test the numerical method, we study analytically a particular class of nonlinear hyperbolic PDEs, deriving closed form solutions for certain special initial data. We also find bounded, smooth, self-similar solutions using group theoretic methods. The numerical method is validated against these analytical results. In addition, we compare the errors in our method with those using the Lax-Wendroff method for both convex and nonconvex flux functions. Finally, we apply the method to solve a PDE with a convex flux function describing the development of a thin liquid film on a horizontally rotating disk and a PDE with a nonconvex flux function, arising in a problem concerning flow in an underground reservoir.