RELAXATION LIMITS FOR A CLASS OF BALANCE LAWS WITH KINETIC FORMULATION

We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.

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A kinetic formulation for scalar balance laws

Transport Theory and Statistical Physics ◽

10.1080/00411459508206029 ◽

1995 ◽

Vol 24 (9) ◽

pp. 1369-1384

Author(s):

Jiaxin Hu

Keyword(s):

Balance Laws ◽

Kinetic Formulation

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Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽

10.3390/math9151799 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1799

Author(s):

Irene Gómez-Bueno ◽

Manuel Jesús Castro Díaz ◽

Carlos Parés ◽

Giovanni Russo

Keyword(s):

High Order ◽

Collocation Methods ◽

Quadrature Formulas ◽

Balance Laws ◽

Source Terms ◽

Time Step ◽

Systems Of Balance Laws ◽

Reconstruction Operator ◽

Non Linear ◽

Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

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