10,129 research outputs found
The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem
AbstractIn modern numerical simulation of prospecting and exploiting oil–gas resources and environmental science, it is important to consider a numerical method for nonlinear convection-dominated diffusion problems. Based on actual conditions, such as the three-dimensional characteristics of large-scale science-engineering computation, we present a kind of characteristic finite volume element method. Some techniques, such as calculus of variations, commutating operators, the theory of prior estimates and techniques, are adopted. Suboptimal order error estimate in L2 norm and optimal order error estimate in H1 norm are derived to determine the errors for the approximate solution. Numerical results are presented to verify the performance of the scheme
A Multiscale Thermo-Fluid Computational Model for a Two-Phase Cooling System
In this paper, we describe a mathematical model and a numerical simulation
method for the condenser component of a novel two-phase thermosyphon cooling
system for power electronics applications. The condenser consists of a set of
roll-bonded vertically mounted fins among which air flows by either natural or
forced convection. In order to deepen the understanding of the mechanisms that
determine the performance of the condenser and to facilitate the further
optimization of its industrial design, a multiscale approach is developed to
reduce as much as possible the complexity of the simulation code while
maintaining reasonable predictive accuracy. To this end, heat diffusion in the
fins and its convective transport in air are modeled as 2D processes while the
flow of the two-phase coolant within the fins is modeled as a 1D network of
pipes. For the numerical solution of the resulting equations, a Dual
Mixed-Finite Volume scheme with Exponential Fitting stabilization is used for
2D heat diffusion and convection while a Primal Mixed Finite Element
discretization method with upwind stabilization is used for the 1D coolant
flow. The mathematical model and the numerical method are validated through
extensive simulations of realistic device structures which prove to be in
excellent agreement with available experimental data
A cut finite element method for coupled bulk-surface problems on time-dependent domains
In this contribution we present a new computational method for coupled
bulk-surface problems on time-dependent domains. The method is based on a
space-time formulation using discontinuous piecewise linear elements in time
and continuous piecewise linear elements in space on a fixed background mesh.
The domain is represented using a piecewise linear level set function on the
background mesh and a cut finite element method is used to discretize the bulk
and surface problems. In the cut finite element method the bilinear forms
associated with the weak formulation of the problem are directly evaluated on
the bulk domain and the surface defined by the level set, essentially using the
restrictions of the piecewise linear functions to the computational domain. In
addition a stabilization term is added to stabilize convection as well as the
resulting algebraic system that is solved in each time step. We show in
numerical examples that the resulting method is accurate and stable and results
in well conditioned algebraic systems independent of the position of the
interface relative to the background mesh
An improved method for solving quasilinear convection diffusion problems on a coarse mesh
A method is developed for solving quasilinear convection diffusion problems
starting on a coarse mesh where the data and solution-dependent coefficients
are unresolved, the problem is unstable and approximation properties do not
hold. The Newton-like iterations of the solver are based on the framework of
regularized pseudo-transient continuation where the proposed time integrator is
a variation on the Newmark strategy, designed to introduce controllable
numerical dissipation and to reduce the fluctuation between the iterates in the
coarse mesh regime where the data is rough and the linearized problems are
badly conditioned and possibly indefinite. An algorithm and updated marking
strategy is presented to produce a stable sequence of iterates as boundary and
internal layers in the data are captured by adaptive mesh partitioning. The
method is suitable for use in an adaptive framework making use of local error
indicators to determine mesh refinement and targeted regularization. Derivation
and q-linear local convergence of the method is established, and numerical
examples demonstrate the theory including the predicted rate of convergence of
the iterations.Comment: 21 pages, 8 figures, 1 tabl
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft
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