Vanishing Viscosity Method Research Articles

Non‐self‐similar solutions containing delta shock waves for nonhomogeneous zero‐pressure gas dynamics with energy conservation law by the viscosity method

This article considers the zero‐pressure gas dynamics governed by the conservation laws of mass, momentum, and energy with time‐dependent source terms. By adopting the vanishing viscosity method, we propose a viscous system to prove the stability of non‐self‐similar Riemann solutions including the delta shock wave and vacuum.

Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the L1 norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of the eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. The implementation of parallel computing using domain decomposition with 1-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code.

Riemann problem for a 2times 2 hyperbolic system with time-gradually-degenerate damping

This paper is focused on the Riemann problem for a 2times 2 hyperbolic system of conservation laws with a time-gradually-degenerate damping. Two kinds of non-self-similar solutions involving the delta-shocks and vacuum are obtained using the variable substitution method. The generalized Rankine-Hugoniot relation and entropy condition are clarified for the delta-shock. Furthermore, the vanishing viscosity method proves the existence, uniqueness, and stability of non-self-similar solutions.

Delta‐shock for a class of systems of conservation laws of the Keyfitz–Kranzer type

AbstractRiemann problem for a class of systems of conservation laws of Keyfitz–Kranzer type is solved. The Riemann solutions contain three kinds of interesting structures, two of which contain vacuum and the other includes delta‐shock. The generalized Rankine–Hugoniot relation and improved entropy condition are proposed to solve the delta‐shock. Besides, with the use of the vanishing viscosity method, all of the existence, uniqueness, and stability of the solutions involving the delta‐shock are proved.

Stochastic fractional conservation laws

In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws with stochastic forcing. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.

Reinforcement learning based optimal synchronization control for multi-agent systems with input constraints using vanishing viscosity method

This paper investigates the optimal synchronization and disturbance rejection problem for leader-follower multi-agent systems (MASs) subject to constrained input using online actor-critic (A-C) algorithm. Different from the existing works on optimal synchronization problems, the common smoothness assumption on the value function used in the typically reinforcement learning (RL) formulations is not satisfied due to the input constraints. To relax this assumption, the vanishing viscosity method is introduced to construct a more general value function which makes the modified Hamilton–Jacobi-Isaacs (HJI) equation admit a smooth solution. Based on the modified HJI equation, an improved A-C neural networks (NNs) algorithm is developed to find a smooth approximation optimal control solution. Finally, a simulation is performed to demonstrate the effectiveness of the proposed approach.

Vanishing viscosity in mean-field optimal control

We show the existence of Lipschitz-in-space optimal controls for a class of mean-field control problems with dynamics given by a non-local continuity equation. The proof relies on a vanishing viscosity method: we prove the convergence of the same problem where a diffusion term is added, with a small viscosity parameter. By using stochastic optimal control, we first show the existence of a sequence of optimal controls for the problem with diffusion. We then build the optimizer of the original problem by letting the viscosity parameter go to zero.

Delta shock wave as limits of vanishing viscosity for zero‐pressure gas dynamics with energy conservation law

AbstractThis paper studies the system of conservation laws of mass, momentum, and energy in zero‐pressure gas dynamics. By the vanishing viscosity method, the stability of the solutions involving delta shock wave with Dirac delta functions developing in both state variables is established.

Applications of wall-models to implicit large eddy simulations in the spectral/hp element method

An implicit large eddy simulation (iLES) method based on a spectral vanishing viscosity (SVV) method has been well established in the framework of the spectral/hp element method (which is called the SVV-iLES method), and the SVV regularization introduces proper dissipation to the numerical scheme, which in turn reduces the truncation error and the uncertainty in computation. However, SVV-iLES is still limited to moderate Reynolds numbers due to the huge computational cost in resolving the inner part of the boundary layer. By modeling the near-wall layer, a wall-model coupling strategy, tailored to the features of the SVV-iLES, is proposed to substantially reduce the computational cost. This work is the first to construct the wall model of SVV-iLES based on the spectral/hp element method, and the practice rules can be regarded as guidelines for the application of the wall model to iLES with high order schemes. Numerical investigations for a well-established benchmark problem of the turbulent channel flow are implemented by the proposed wall-modeled SVV-iLES. The results obtained are consistent with the reference direct numerical simulation (DNS) data at different Reynolds numbers considered. The effect of the time-averaged velocity at the matching interface is examined, and the results suggest that the instantaneous velocity can be directly used in the algebraic wall-model, which in turn could speed up the computation. Additionally, a comparison between the classical log-law and Reichardt's law models reveals that there are no significant differences in statistical quantities in the logarithmic region. Finally, it was found that, at similar grid resolution, the p-type refinement shows better agreement with the reference DNS results.

Viscosity solutions for doubly nonlinear evolution equations

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

Construction of approximate solutions to the riemann problem for two-phase flow of immiscible liquids by modifying the vanishing viscosity method

In the paper, we apply the vanishing viscosity method for an approximate solution to the Riemann problem. This approach gives the effects of the accuracy of the solution and the speed of convergence by discrediting time and spatial variables. The obtained method ensures the smoothness of the solution without taking into account the capillary pressure. The results confirm the negligible influence of cross-link conditions compared to the classical Darcy approach. The proposed solutions of the new approach are intended to improve the methods and schemes of discretization both in space and in time. This is achieved by minimizing viscosity, and discretization in space and time. These factors are of paramount importance for studying phenomena with variable saturation in the transient mode and analyzing water/oil flows and migrations in real time, since discretization in space and time affects the accuracy and convergence of calculations. Our result in the form of obtaining viscous solutions of the filtration process is interesting from a theoretical point of view. From a practical point of view, numerical modeling allows early prediction of performance. Thus, the applied aspect of using the obtained scientific result is the possibility of improving the process by taking into account the influence of phases of fluid flows

Path-Dependent Hamilton--Jacobi Equations with Super-Quadratic Growth in the Gradient and the Vanishing Viscosity Method

The nonexponential Schilder-type theorem in Backhoff-Veraguas, Lacker, and Tangpi [Ann.\Appl. Probab., 30 (2020), pp. 1321--1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton--Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton--Jacobi--Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semicontinuous solutions holds, and state constraints are admitted.

The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations

We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

Global Entropy Solutions and Newtonian Limit for the Relativistic Euler Equations

We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on a careful entropy analysis, we establish the compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.

Efficient SVV stabilized triangular spectral element methods for incompressible flows of high Reynolds numbers

In this paper, we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows. This can be regarded as an extension of a similar stabilization technique for the standard spectral element method. The difficulty of this extension lies in the fact that a suitable definition of spectral vanishing viscosity operator in non-structured elements does not exist, and it is not clear that if a suitably defined spectral vanishing viscosity provides desirable dissipation for the artificially accumulated energy. The main contribution of the paper includes: 1) a well-defined spectral vanishing viscosity operator is proposed for non-standard spectral element methods for the Navier-Stokes equations based on triangular or tetrahedron partitions; 2) an evaluation technique is introduced to efficiently implement the stabilization term without extra computational cost; 3) the accuracy and efficiency of the proposed method is carefully examined through several numerical examples. Our numerical results show that the proposed method not only preserves the exponential convergence, but also produces improved accuracy when applied to the unsteady Navier-Stokes equations having smooth solutions. Especially, the stabilized triangular spectral element method efficiently stabilizes the simulation of high Reynolds incompressible flows.

The strong trace property and the Neumann problem for stochastic conservation laws

We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a crucial point in order to prove the uniqueness of the kinetic solution to the referred problem we establish a new strong trace theorem for stochastic conservation laws, which extends to the stochastic context the pioneering theorem by Vasseur. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.

Selection by vanishing common noise for potential finite state mean field games

The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98–162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality.

An hyperbolic-parabolic predator-prey model involving a vole population structured in age

We prove existence and stability of entropy solutions for a predator-prey system consisting of an hyperbolic equation for predators and a parabolic-hyperbolic equation for preys. The preys' equation, which represents the evolution of a population of voles as in [2], depends on time, t, age, a, and on a 2-dimensional space variable x, and it is supplemented by a nonlocal boundary condition at a=0. The drift term in the predators' equation depends nonlocally on the density of preys and the two equations are also coupled via classical source terms of Lotka-Volterra type, as in [4]. We establish existence of solutions by applying the vanishing viscosity method, and we prove stability by a doubling of variables type argument.

Uniqueness of dissipative solution for Camassa–Holm equation with peakon–antipeakon initial data

We give a proof for the uniqueness of dissipative solution for Camassa–Holm equation with some peakon–antipeakon initial data. Our result shows that two existing global existence frameworks, through the vanishing viscosity method by Xin–Zhang in Xin and Zhang (2000) and the transformation of coordinate method for dissipative solutions by Bressan–Constantin in Bressan and Constantin (2007), give the same solution, for a special but typical initial data forming finite time gradient blowup.

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method “à la Kružkov” for more general equations where the noise coefficient may depend explicitly on the spatial variable.