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| Paper IPM / M / 7374 |
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| Abstract: | |
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Numerical techniques based on finite difference schemes leading to
parallel algorithms have been developed for obtaining approximate
solutions to an initial-boundary value problem for the
two-dimensional parabolic partial differential equation (PDE) with
a non-linear boundary condition. This class of parabolic PDEs
plays a very important role in many branches of science and
engineering. The non-linear condition is in the form of a double
integral giving the specification of mass in the solution domain.
Not only the problem has both Neumann and Dirichlet boundary
conditions but the Dirichlet boundary condition is in a
non-standard form. While sharing some common features with the
one-dimensional models, the solution of two-dimensional equations
are substantially more difficult, thus some considerations are
taken to be able to extend some ideas of one-dimensional case. Due
to the structure of the boundary conditions a reduced
one-dimensional parabolic equation for the new unknown
v(y,t)=∫10 u(x,y,t)dx can be formulated. The resulting
problem has a non-local boundary condition. The new
two-dimensional parabolic PDE with Neumann's boundary conditions
will be solved numerically by using the method of lines
semi-discretization approach. The space derivatives in the PDE are
approximated by finite difference replacements. The solution of
the resulting system of first-order linear ordinary differential
equations satisfies a recurrence relation which involves a matrix
exponential function. The accuracy in time is controlled by
choosing several subdiagonal Pade approximants to replace this
matrix exponential term. Numerical techniques are developed to
compute the required solution using a splitting method, leading to
algorithms for sequential and parallel implementation. The
algorithms are tested on a model problem from the literature. The
article concludes with the results of some numerical experiments.
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