Weak Asymptotics Method Research Articles

Semi-discrete Lagrangian–Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions

In this work, we have expanded upon the (local) semi-discrete Lagrangian–Eulerian method initially introduced in Abreu et al. (2022) to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux, referred to as the nonlinear nonlocal model: ∂tρ(t,x)+∑i=1d∂xi(Vi[W[ρ,ω](t,x)]Fi(ρ(t,x)))=0,(t,x)∈(0,T)×Rd. For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in Abreu et al. (2016), with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian–Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space L1(Rd)∩L∞(Rd). 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the L1-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution of Eq. (1). Finally, we present a section of numerical examples to illustrate our results. In particular, we have examined examples discussed in Aggarwal et al. (2015) and Keimer et al. (2018). In addition, we also provide numerical results for a nonlocal impact of the form ωη∗ρ, where η=0.1 for class of the two-dimensional nonlinear nonlocal inviscid Burgers’ equations.

$$\delta '''$$-shock wave solution to a nonstrictly hyperbolic system of conservation laws using weak asymptotic method

$$\delta '''$$-shock wave solution to a nonstrictly hyperbolic system of conservation laws using weak asymptotic method

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws.

Interaction of δ-Shock Waves in a System of Pressureless Gas Dynamics Equations

In the present paper, an analytic description of nonlinear delta-shock waves interaction is described by using the weak asymptotics method.

Stepwise Approximations and Limiting Generalized Solution to The Hyperbolic Conservation Law

In this paper we present a modification of the well-known front-tracking method. This approach is based on the use of both stable and unstable jumps which finally produce an admissible solution. Instead of the entropy condition, we use the continuous-in-time description of the jupm interaction to select ”right” solutions. To do this we use a weak asymptotics method developed in part by the author.

The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product

The system of conservation laws \({u_t} + {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)_x} = 0\), vt + (uv − v)x = 0 with the initial conditions u(x, 0) = l0 + b0H(x), v(x, 0) = k0 + c0H(x), where H is the Heaviside function is studied. This strictly hyperbolic system was introduced by M. Brio in 1988 and provides a simplified model for the magnetohydrodynamics equations. Under certain compatibility conditions for the constants l0, b0, k0, c0, an explicit solution containing a Dirac mass is given and we prove the uniqueness of this solution within a convenient class of distributions which includes Dirac-delta measures. Our concept of solution is defined within the framework of a distributional product, and it is a consistent extension of the concept of a classical solution. This direct method seems considerably simpler than the weak asymptotic method usually used in the study of delta-shocks emergence in nonlinear conservation laws.

Weak asymptotic methods for 3-D self-gravitating pressureless fluids. Application to the creation and evolution of solar systems from the fully nonlinear Euler-Poisson equations

We construct a family of classical continuous functions S(x, y, z, t, ϵ) which tend to satisfy asymptotically the system of selfgravitating pressureless fluids when ϵ → 0. This produces a weak asymptotic method in the sense of Danilov, Omel’yanov, and Shelkovich. The construction is based on a family of two ordinary differential equations (ODEs) (one for the continuity equation and one for the Euler equation) in classical Banach spaces of continuous functions. This construction applies to 3-D self-gravitating pressureless fluids even in presence of point and string concentrations of matter. The method is constructive which permits to check numerically from standard methods for ODEs that these functions tend to the known or admitted solutions when the latter exist. As a direct application, we present a simulation of formation and evolution of a planetary system from a rotating disk of dust: a theorem in this paper asserts that the observed results are a depiction of functions that satisfy the system with arbitrary precision.

Non-standard shocks in the Buckley–Leverett equation

It is shown how delta shock waves which consist of Dirac delta distributions and classical shocks can be used to construct non-monotone solutions of the Buckley–Leverett equation. These solutions are interpreted using a recent variational definition of delta shock waves in which the Rankine–Hugoniot deficit is explicitly accounted for [6]. The delta shock waves are also limits of approximate solutions constructed using a recent extension of the weak asymptotic method to complex-valued approximations [15]. Finally, it is shown how these non-standard shocks can be fitted together to construct similarity and traveling-wave solutions which are non-monotone, but still admissible in the sense that characteristics either enter or are parallel to the shock trajectories.

Singular solutions of a fully nonlinear 2 × 2 system of conservation laws

AbstractExistence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyperbolic system of equationsThe system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

Shock wave formation process for a multidimensional scalar conservation law

We construct a global smooth approximate solution to a multidimensional scalar conservation law describing the shock wave formation process for initial data with small variation. In order to solve the problem, we modify the method of character- istics by introducing new characteristics | nonintersecting curves along which the (approximate) solution to the problem under study is constant. The procedure is based on the weak asymptotic method, a technique which appeared to be rather powerful for investigating nonlinear waves interactions

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

AbstractUsing the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics.Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.

Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.

Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

Weak asymptotic solution of the phase-field system in the case of fusion of free boundaries in the Stefan-Gibbs-Thomson problem

The process of fusion of free boundaries in binary media in the Stefan-Gibbs-Thomson problem is analyzed. By using the weak asymptotics method, the formula for the weak asymptotic solution of the phase-field system global in time is constructed in the case considered.

Delta-shock wave type solution of hyperbolic systems of conservation laws

For two classes of hyperbolic systems of conservation laws new definitions of a δ \delta -shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of δ \delta -shocks. The weak asymptotics method developed by the authors is used to describe the propagation of δ \delta -shock waves to the three types of systems of conservation laws and derive the corresponding Rankine–Hugoniot conditions for δ \delta -shocks.

Dynamics of propagation and interaction of [formula omitted]-shock waves in conservation law systems

We introduce a new definition of a δ -shock wave type solution for a class of systems of conservation laws in the one-dimensional case. The weak asymptotics method developed by the authors is used to construct formulas describing the propagation and interaction of δ -shock waves. The dynamics of merging two δ -shocks is described by explicit formulas continuously in time.

Weak asymptotics method and the interaction of infinitely narrow δ-solitons

We present a new method for studying the interaction of solitons for nonintegrable KdV-type equations with small dispersion and test this method for the KdV equation.