A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
A high-order compact finite difference scheme for a fully nonlinear
parabolic differential equation is analyzed. The equation arises in the
modeling of option prices in financial markets with transaction costs.
It is shown that the finite difference solution converges locally
uniformly to the unique viscosity solution of the continuous equation.
The proof is based on a careful study of the discretization matrices and on
an abstract convergence result due to Barles and Souganides.
Tuning the alternating Schwarz method to the
exterior problems is the subject of this paper.
We present the original algorithm
and we propose a modification of it, so that the
solution of the subproblem involving the condition at infinity
has an explicit integral representation formulas while the solution
of the other subproblem, set in a bounded domain,
is approximated by classical variational methods.
We investigate many of the advantages of the new
Schwarz approach: a geometrical convergence...
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