Central finite volume schemes with constrained transport divergence treatment for three-dimensional ideal MHD

2006 ◽  
Vol 212 (2) ◽  
pp. 617-636 ◽  
Author(s):  
R. Touma ◽  
P. Arminjon
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Finite Volume Schemes ◽  
Ideal Mhd ◽  
Constrained Transport

scholarly journals Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method

2018 ◽  
Vol 6 (6) ◽  
pp. 107-123
Author(s):  
Yaw Kyei
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Integral Form ◽  
Higher Order ◽  
Finite Volume Schemes ◽  
Finite Volume Discretization ◽  
Equation Error ◽  
Error Expansion ◽  
Control Volumes ◽  
Physical Phenomena

Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

CONVERGENCE OF A FINITE VOLUME SCHEME FOR GAS–WATER FLOW IN A MULTI-DIMENSIONAL POROUS MEDIUM

2013 ◽  
Vol 24 (01) ◽  
pp. 145-185 ◽  
Author(s):  
MOSTAFA BENDAHMANE ◽  
ZIAD KHALIL ◽  
MAZEN SAAD
Keyword(s):  
Porous Medium ◽  
Finite Volume ◽  
Three Dimensional ◽  
Energy Estimates ◽  
Finite Volume Scheme ◽  
Convergence Properties ◽  
Finite Volume Schemes ◽  
Finite Element Scheme ◽  
Discrete Energy ◽  
Volume Scheme

This paper deals with construction and convergence analysis of a finite volume scheme for compressible/incompressible (gas–water) flows in porous media. The convergence properties of finite volume schemes or finite element scheme are only known for incompressible fluids. We present a new result of convergence in a two or three dimensional porous medium and under the only consideration that the density of gas depends on global pressure. In comparison with incompressible fluid, compressible fluids requires more powerful techniques; especially the discrete energy estimates are not standard.

scholarly journals Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

2011 ◽  
Vol 9 (2) ◽  
pp. 324-362 ◽  
Author(s):  
Franz Georg Fuchs ◽  
Andrew D. McMurry ◽  
Siddhartha Mishra ◽  
Nils Henrik Risebro ◽  
Knut Waagan
Keyword(s):  
Finite Volume ◽  
Source Term ◽  
Second Order ◽  
High Order ◽  
Mhd Equations ◽  
Positive Pressure ◽  
Finite Volume Schemes ◽  
Riemann Solvers ◽  
Ideal Mhd ◽  
Approximate Riemann Solvers

AbstractWe design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on very fine meshes.

Splitting based finite volume schemes for ideal MHD equations

2009 ◽  
Vol 228 (3) ◽  
pp. 641-660 ◽  
Author(s):  
F.G. Fuchs ◽  
S. Mishra ◽  
N.H. Risebro
Keyword(s):  
Finite Volume ◽  
Mhd Equations ◽  
Finite Volume Schemes ◽  
Ideal Mhd

scholarly journals Comparison of finite-volume schemes for diffusion problems

2018 ◽  
Vol 73 ◽  
pp. 82 ◽  
Author(s):  
Martin Schneider ◽  
Dennis Gläser ◽  
Bernd Flemisch ◽  
Rainer Helmig
Keyword(s):  
Finite Volume ◽  
Approximation Scheme ◽  
Model Problem ◽  
Three Dimensional ◽  
Maximum Principles ◽  
Test Cases ◽  
Finite Volume Schemes ◽  
Flux Approximation ◽  
Elliptic Model ◽  
Diffusion Problems

We present an abstract discretization framework and demonstrate that various cell-centered and hybrid finite-volume schemes fit into it. The different schemes considered in this work are then analyzed numerically for an elliptic model problem with respect to the properties consistency, coercivity, extremum principles, and sparsity. The test cases presented comprise of two- and three-dimensional setups, mildly and highly anisotropic tensors and grids of different complexities. The results show that all schemes show a similar convergence behavior, except for the two-point flux approximation scheme, and seem to be coercive. Furthermore, they confirm that linear schemes, in contrast to nonlinear schemes, are in general neither positivity-preserving nor satisfy discrete minimum or maximum principles.

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