Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes

2021 ◽  
Vol 30 (4) ◽  
pp. 1185-1215
Author(s):  
global sci
Keyword(s):  
Heat Conduction ◽  
Finite Volume ◽  
Three Dimensional ◽  
Finite Volume Schemes ◽  
Positivity Preserving ◽  
Polyhedral Meshes

scholarly journals Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

2011 ◽  
Vol 467 (2134) ◽  
pp. 2752-2776 ◽  
Author(s):  
Xiangxiong Zhang ◽  
Chi-Wang Shu
Keyword(s):  
Maximum Principle ◽  
Conservation Laws ◽  
Finite Volume ◽  
Computational Cost ◽  
High Order ◽  
High Order Accuracy ◽  
Boltzmann Transport ◽  
Finite Volume Schemes ◽  
Positivity Preserving ◽  
High Order Schemes

In an earlier study (Zhang & Shu 2010 b J. Comput. Phys. 229 , 3091–3120 ( doi:10.1016/j.jcp.2009.12.030 )), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov–Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

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 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.

scholarly journals Conservative Finite Volume Schemes for Multidimensional Fragmentation Problems

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 635
Author(s):  
Jitraj Saha ◽  
Andreas Bück
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Test Problems ◽  
Finite Volume Scheme ◽  
Finite Volume Schemes ◽  
One Dimensional ◽  
Volume Scheme ◽  
Multidimensional Problems ◽  
Conservative Form ◽  
Fragmentation Equation

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems.

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