scholarly journals Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

2008 ◽  
Vol 42 (5) ◽  
pp. 699-727 ◽  
Author(s):  
Sébastien Martin ◽  
Julien Vovelle
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Scalar Conservation Laws ◽  
Flux Function ◽  
Discontinuous Flux Function ◽  
Discontinuous Flux ◽  
Scalar Conservation

scholarly journals High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1079
Author(s):  
Tingting Xiang ◽  
Guodong Wang ◽  
Suping Zhang
Keyword(s):  
Conservation Laws ◽  
High Order ◽  
Splitting Scheme ◽  
Scalar Conservation Laws ◽  
Flux Function ◽  
Flux Splitting ◽  
Discontinuous Flux Function ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Accurate Scheme

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

scholarly journals Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

2009 ◽  
Vol 2009 ◽  
pp. 1-33 ◽  
Author(s):  
H. Holden ◽  
K. H. Karlsen ◽  
D. Mitrovic
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Flux Function ◽  
Discontinuous Flux Function ◽  
Scalar Conservation Law ◽  
Discontinuous Flux ◽  
Dispersion Terms ◽  
Scalar Conservation

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach ofH-measures to investigate the zero diffusion-dispersion-smoothing limit.

scholarly journals Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux

2021 ◽  
Author(s):  
Jayesh Badwaik ◽  
Christian Klingenberg ◽  
Nils Henrik Risebro ◽  
Adrian M Ruf
Keyword(s):  
Monte Carlo ◽  
Conservation Laws ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Spatial Dependency ◽  
Multilevel Monte Carlo ◽  
Two Phase ◽  
Flux Function ◽  
Discontinuous Flux ◽  
Volume Approximation

We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.

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