BAIL 2006 Book of Abstracts - Institut für Numerische und ...

S. LI, L.P. SHISHKINA, G.I. SHISHKIN: Parameter-Uniform Method for a Singularly
Perturbed Parabolic Equation Modelling the Black-Scholes equation in the Presence
of Interior and Boundary Layers
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Parameter–Uniform Method for a Singularly Perturbed Parabolic Equation
Modelling the Black–Scholes equation in the Presence of Interior and
Boundary Layers ∗
Shuiying Li 1 , Lidia P. Shishkina 2 and Grigory I. Shishkin 2
1 Department of Computational Science,
National University of Singapore,
Singapore 117543
scip1075@nus.edu.sg
2 Institute of Mathematics and Mechanics,
Ural Branch of Russian Academy of Sciences,
Yekaterinburg 620219, Russia
Lida@convex.ru and shishkin@imm.uran.ru
Solutions of regular parabolic equations with nonsmooth initial data are of limited smoothness
themselves (in a neighbourhood of the nonsmoothness of the data). When the equation
is singularly perturbed, its solution has singularities such as initial, boundary and/or interior
layers with own scales. These problems belong to the class of singularly perturbed multiscale
problems. The multiscale behaviour of solutions complicates the construction and study of
efficient patameter-uniform numerical methods.
A singularly perturbed parabolic convection-diffusion equation with the second derivative
multiplied by a singular perturbation parameter ε, which ranges in (0, 1], arises when we model
the Black-Scholes equation upon a European call option (see, e.g., [1]). There are a few singularities
in the problem, i.e., a single discontinuity of the first derivative of the initial condition
at a point x0, the unbounded growth of the initial condition at infinity and the unbounded
domain in space. The solution of this initial value problem grows exponentially without bound
as x → ∞. Thus, we deal with a singularly perturbed multiscale problem that has different
types of singularities.
In this presentation we consider a boundary value problem (in a bounded domain) for this
singularly perturbed parabolic convection-diffusion equation with an additional singularity generated
by the discontinuity of the first derivative of the initial function. For such a problem,
singularities such as a boundary and an interior layer with own specific scales arise. For small
values of the parameter ε, the interior layer due to nonsmooth initial data appears in a neighbourhood
of the characteristic of the reduced equation passing through the point (x0, 0), and
the regular boundary layer appears in a neighbourhood of the outflow boundary through which
the convective flux leaves the domain.
By using the singularity splitting method and piecewise uniform meshes condensing in the
boundary layer, we construct a special difference scheme that allows us to approximate εuniformly
the solution of the problem under consideration, as well as the first derivative of
the solution. The corresponding theoretical and numerical results are provided in the paper.
The description of the technique applied in this study can be found in [2]. Preliminary results
for a similar problem with no boundary layer were given in [3].
∗ This research was supported in part by the Russian Foundation for Basic Research (grant No 04-01-00578,
04–01–89007–NWO a) and by the Dutch Research Organization NWO under grant No 047.016.008.
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Speaker: SHISHKINA, L.P. 59 BAIL 2006
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