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A family of difference schemes for fourth order parabolic partial differential equations
A family of methods is developed for the numerical solution of fourth
order parabolic partial differential equations in one- and two-space
variables. The methods are seen to evolve from multiderivative methods
for second order ordinary differential equations.
The methods are tested on three model problems, with constant
coefficients and variable coefficients, which have appeared in the literature
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Global extrapolation procedures for linear partial differential equations
Global extrapolation procedures, in space and time are considered for the numerical Solution of linear partial differential equations. Global extrapolation procedures in time only are reviewed.
The procedures are tested on three problems from the literature, one of which has a nonlinear source term
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
Numerical Methods for a Nonlinear BVP Arising in Physical Oceanography
In this paper we report and compare the numerical results for an ocean
circulation model obtained by the classical truncated boundary formulation, the
free boundary approach and a quasi-uniform grid treatment of the problem. We
apply a shooting method to the truncated boundary formulation and finite
difference methods to both the free boundary approach and the quasi-uniform
grid treatment. Using the shooting method, supplemented by the Newton's
iterations, we show that the ocean circulation model cannot be considered as a
simple test case. In fact, for this method we are forced to use as initial
iterate a value close to the correct missing initial condition in order to be
able to get a convergent numerical solution. The reported numerical results
allow us to point out how the finite difference method with a quasi-uniform
grid is the less demanding approach and that the free boundary approach
provides a more reliable formulation than the classical truncated boundary
formulation.Comment: 25 pages, 12 figures, 5 table
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