103,891 research outputs found
Finite Volume Transport Schemes
We analyze arbitrary order linear nite volume transport schemes and show asymptotic stability in L1 and L1 for odd order schemes in dimen- sion one. It gives sharp fractional order estimates of convergence for BV solutions. It shows odd order nite volume advection schemes are better than even order nite volume schemes. Therefore the Gibbs phenomena is controled for odd order nite volume schemes. Numerical experiments sustain the theoretical analysis. In particular the oscillations of the Lax- Wendro scheme for small Courant numbers are correlated with its non stability in L1. A scheme of order three is proved to be stable in L1 and L1
Numerical Methods for Solving Convection-Diffusion Problems
Convection-diffusion equations provide the basis for describing heat and mass
transfer phenomena as well as processes of continuum mechanics. To handle flows
in porous media, the fundamental issue is to model correctly the convective
transport of individual phases. Moreover, for compressible media, the pressure
equation itself is just a time-dependent convection-diffusion equation.
For different problems, a convection-diffusion equation may be be written in
various forms. The most popular formulation of convective transport employs the
divergent (conservative) form. In some cases, the nondivergent (characteristic)
form seems to be preferable. The so-called skew-symmetric form of convective
transport operators that is the half-sum of the operators in the divergent and
nondivergent forms is of great interest in some applications.
Here we discuss the basic classes of discretization in space: finite
difference schemes on rectangular grids, approximations on general polyhedra
(the finite volume method), and finite element procedures. The key properties
of discrete operators are studied for convective and diffusive transport. We
emphasize the problems of constructing approximations for convection and
diffusion operators that satisfy the maximum principle at the discrete level
--- they are called monotone approximations.
Two- and three-level schemes are investigated for transient problems.
Unconditionally stable explicit-implicit schemes are developed for
convection-diffusion problems. Stability conditions are obtained both in
finite-dimensional Hilbert spaces and in Banach spaces depending on the form in
which the convection-diffusion equation is written
Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations
This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook)
schemes in the framework of finite volume method for the ultra-relativistic
flows. Different from the existing kinetic flux-vector splitting (KFVS) or
BGK-type schemes for the ultra-relativistic Euler equations, the present
genuine BGK schemes are derived from the analytical solution of the
Anderson-Witting model, which is given for the first time and includes the
"genuine" particle collisions in the gas transport process. The BGK schemes for
the ultra-relativistic viscous flows are also developed and two examples of
ultra-relativistic viscous flow are designed. Several 1D and 2D numerical
experiments are conducted to demonstrate that the proposed BGK schemes not only
are accurate and stable in simulating ultra-relativistic inviscid and viscous
flows, but also have higher resolution at the contact discontinuity than the
KFVS or BGK-type schemes.Comment: 41 pages, 13 figure
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
Contributions to the numerical solution of heterogeneous fluid mechanics models
A high order projection hybrid finite volume â finite element method is developed to solve incompressible and compressible low Mach number flows. Furthermore, turbulent regimes are also considered thanks to the kâΔ model. The unidimensional advection-diffusion-reaction equation is used to construct, analyze and assess high order finite volume schemes. Two families of methods are studied: Kolgan-type schemes and ADER methodology. A modification of the last one is proposed providing a new numerical method called Local ADER. The designed method is extended to solve the transport-diffusion stage of the three-dimensional projection method. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the schemes and to assess the performance of the method on several realistic test problems
Dynamical Core Model Intercomparison Project:Tracer Transport Test Cases
Threeâdimensional advection tests are required to assess the ability of transport schemes of dynamical cores to model tracer transport on the sphere accurately. A set of three tracerâtransport test cases for threeâdimensional flow is presented. The tests focus on the physical and numerical issues that are relevant to threeâdimensional tracer transport: positivity preservation, interâtracer correlations, horizontalâvertical coupling, order of accuracy and choice of vertical coordinate. The first test is a threeâdimensional deformational flow. The second test is a Hadleyâlike global circulation. The final test is a solidâbody rotation test in the presence of rapidly varying orography. A variety of assessment metrics, such as error norms, convergence rates and mixing diagnostics, are used. The tests are designed for easy implementation within existing and developing dynamical cores and have been a cornerstone of the 2012 Dynamical Core Model Intercomparison Project ( DCMIP ). Example results are shown using the transport schemes in two dynamical cores: the Community Atmosphere Model finiteâvolume dynamical core ( CAMâFV ) and the cubedâsphere finiteâvolume MCore dynamical core.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/107545/1/qj2208.pd
Development of high-order realizable finite-volume schemes for quadrature-based moment method
Kinetic equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of velocity distribution. Recently, a quadrature-based moment method was derived by Fox for approximating solutions to kinetic equation for arbitrary Knudsen number. Fox also described 1st- and 2nd-order finite-volume schemes for solving the equations. The success of the new method is based on a moment-inversion algorithm that is used to calculate non-negative weights and abscissas from moments. The moment-inversion algorithm does not work if the moments are non-realizable, meaning they do not correspond to a distribution function. Not all the finite-volume schemes lead to realizable moments. Desjardins et al. showed that realizability is guaranteed with the 1 st-order finite-volume scheme, but at the expense of excess numerical diffusion. In the present work, the nonrealizability of the standard 2 nd-order finite-volume scheme is demonstrated and a generalized idea for the development of high-order realizable finite-volume schemes for quadrature-based moment methods is presented. This marks a significant improvement in the accuracy of solutions using the quadrature-based moment method as the use of 1st-order scheme to guarantee realizability is no longer a limitation
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