248 research outputs found
Piecewise linear transformation in diffusive flux discretization
To ensure the discrete maximum principle or solution positivity in finite
volume schemes, diffusive flux is sometimes discretized as a conical
combination of finite differences. Such a combination may be impossible to
construct along material discontinuities using only cell concentration values.
This is often resolved by introducing auxiliary node, edge, or face
concentration values that are explicitly interpolated from the surrounding cell
concentrations. We propose to discretize the diffusive flux after applying a
local piecewise linear coordinate transformation that effectively removes the
discontinuities. The resulting scheme does not need any auxiliary
concentrations and is therefore remarkably simpler, while being second-order
accurate under the assumption that the structure of the domain is locally
layered.Comment: 11 pages, 1 figures, preprint submitted to Journal of Computational
Physic
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, and the
diffusivity coefficient to be a second-order symmetric and positive definite
tensor. It is well-known that this particular equation is a second-order
elliptic equation, and satisfies a maximum principle under certain regularity
assumptions. However, the finite element implementation of the classical
Galerkin formulation for both anisotropic and isotropic diffusion with decay
does not respect the maximum principle.
We first show that the numerical accuracy of the classical Galerkin
formulation deteriorates dramatically with increase in the decay coefficient
for isotropic medium and violates the discrete maximum principle. However, in
the case of isotropic medium, the extent of violation decreases with mesh
refinement. We then show that, in the case of anisotropic medium, the classical
Galerkin formulation for anisotropic diffusion with decay violates the discrete
maximum principle even at lower values of decay coefficient and does not vanish
with mesh refinement. We then present a methodology for enforcing maximum
principles under the classical Galerkin formulation for anisotropic diffusion
with decay on general computational grids using optimization techniques.
Representative numerical results (which take into account anisotropy and
heterogeneity) are presented to illustrate the performance of the proposed
formulation
A mimetic finite difference based quasi-static magnetohydrodynamic solver for force-free plasmas in tokamak disruptions
Force-free plasmas are a good approximation where the plasma pressure is tiny
compared with the magnetic pressure, which is the case during the cold vertical
displacement event (VDE) of a major disruption in a tokamak. On time scales
long compared with the transit time of Alfven waves, the evolution of a
force-free plasma is most efficiently described by the quasi-static
magnetohydrodynamic (MHD) model, which ignores the plasma inertia. Here we
consider a regularized quasi-static MHD model for force-free plasmas in tokamak
disruptions and propose a mimetic finite difference (MFD) algorithm. The full
geometry of an ITER-like tokamak reactor is treated, with a blanket module
region, a vacuum vessel region, and the plasma region. Specifically, we develop
a parallel, fully implicit, and scalable MFD solver based on PETSc and its
DMStag data structure for the discretization of the five-field quasi-static
perpendicular plasma dynamics model on a 3D structured mesh. The MFD spatial
discretization is coupled with a fully implicit DIRK scheme. The algorithm
exactly preserves the divergence-free condition of the magnetic field under the
resistive Ohm's law. The preconditioner employed is a four-level fieldsplit
preconditioner, which is created by combining separate preconditioners for
individual fields, that calls multigrid or direct solvers for sub-blocks or
exact factorization on the separate fields. The numerical results confirm the
divergence-free constraint is strongly satisfied and demonstrate the
performance of the fieldsplit preconditioner and overall algorithm. The
simulation of ITER VDE cases over the actual plasma current diffusion time is
also presented.Comment: 43 page
An Efficient Method For Solving Highly Anisotropic Elliptic Equations
Solving elliptic PDEs in more than one dimension can be a computationally
expensive task. For some applications characterised by a high degree of
anisotropy in the coefficients of the elliptic operator, such that the term
with the highest derivative in one direction is much larger than the terms in
the remaining directions, the discretized elliptic operator often has a very
large condition number - taking the solution even further out of reach using
traditional methods. This paper will demonstrate a solution method for such
ill-behaved problems. The high condition number of the D-dimensional
discretized elliptic operator will be exploited to split the problem into a
series of well-behaved one and (D-1)-dimensional elliptic problems. This
solution technique can be used alone on sufficiently coarse grids, or in
conjunction with standard iterative methods, such as Conjugate Gradient, to
substantially reduce the number of iterations needed to solve the problem to a
specified accuracy. The solution is formulated analytically for a generic
anisotropic problem using arbitrary coordinates, hopefully bringing this method
into the scope of a wide variety of applications.Comment: 37 pages, 11 figure
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
A Comparison of Consistent Discretizations for Elliptic Problems on Polyhedral Grids
In this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.acceptedVersio
A Fast Semi-implicit Method for Anisotropic Diffusion
Simple finite differencing of the anisotropic diffusion equation, where
diffusion is only along a given direction, does not ensure that the numerically
calculated heat fluxes are in the correct direction. This can lead to negative
temperatures for the anisotropic thermal diffusion equation. In a previous
paper we proposed a monotonicity-preserving explicit method which uses limiters
(analogous to those used in the solution of hyperbolic equations) to
interpolate the temperature gradients at cell faces. However, being explicit,
this method was limited by a restrictive Courant-Friedrichs-Lewy (CFL)
stability timestep. Here we propose a fast, conservative, directionally-split,
semi-implicit method which is second order accurate in space, is stable for
large timesteps, and is easy to implement in parallel. Although not strictly
monotonicity-preserving, our method gives only small amplitude temperature
oscillations at large temperature gradients, and the oscillations are damped in
time. With numerical experiments we show that our semi-implicit method can
achieve large speed-ups compared to the explicit method, without seriously
violating the monotonicity constraint. This method can also be applied to
isotropic diffusion, both on regular and distorted meshes.Comment: accepted in the Journal of Computational Physics; 13 pages, 7
figures; updated to the accepted versio
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