scholarly journals Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

2012 ◽  
Vol 231 (8) ◽  
pp. 3365-3388 ◽  
Author(s):  
S. Mishra ◽  
Ch. Schwab ◽  
J. Šukys
Keyword(s):  
Monte Carlo ◽  
Nonlinear Systems ◽  
Conservation Laws ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Systems Of Conservation Laws ◽  
Multi Level

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

2017 ◽  
Vol 33 (1) ◽  
pp. 25-52 ◽  
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.

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.

Sign in / Sign up

Export Citation Format

Share Document