Stable and accurate hybrid finite volume methods based on pure convexity arguments for hyperbolic systems of conservation law

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

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High-order IMEX-RK finite volume methods for multidimensional hyperbolic systems

Numerical Mathematics and Advanced Applications ◽

10.1007/978-88-470-2089-4_3 ◽

2003 ◽

pp. 35-44

Author(s):

E. Bertolazzi ◽

G. Manzini

Keyword(s):

Finite Volume ◽

Hyperbolic Systems ◽

Finite Volume Methods ◽

High Order

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On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws

The International Journal of High Performance Computing Applications ◽

10.1177/1094342017691876 ◽

2017 ◽

Vol 33 (1) ◽

pp. 25-52 ◽

Cited By ~ 3

Author(s):

J Loffeld ◽

JAF Hittinger

Keyword(s):

Conservation Laws ◽

Finite Volume ◽

Hyperbolic Systems ◽

Finite Volume Methods ◽

High Order ◽

Floating Point ◽

Systems Of Conservation Laws ◽

Data Transfers ◽

Blue Gene ◽

Theory And Experiments

It has been conjectured that higher-order discretizations for partial differential equations will have advantages over the lower-order counterparts commonly used today. The reasoning is that the increase in arithmetic operations will be more than offset by the reduction in data transfers and the increase in concurrent floating-point units. To evaluate this conjecture, the arithmetic intensity of a class of high-order finite-volume discretizations for hyperbolic systems of conservation laws is theoretically analyzed for spatial discretizations from orders three through eight in arbitrary dimensions. Three cache models are considered: the limiting cases of no cache and infinite cache as well as a finite-sized cache model. Models are validated experimentally by measuring floating-point operations and data transfers on an IBM Blue Gene/Q node. Theory and experiments demonstrate that high-order finite-volume methods will be able to provide increases in arithmetic intensity that will be necessary to make better utilization of on-node floating-point capability.

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Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202520500141 ◽

2020 ◽

Vol 30 (03) ◽

pp. 539-609 ◽

Cited By ~ 1

Author(s):

U. S. Fjordholm ◽

K. Lye ◽

S. Mishra ◽

F. Weber

Keyword(s):

Conservation Laws ◽

Finite Volume ◽

Hyperbolic Systems ◽

Global Solutions ◽

Monte Carlo Sampling ◽

Finite Volume Methods ◽

Sampling Procedure ◽

Convergence Theory ◽

Statistical Solution ◽

Statistical Solutions

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.

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