Numerical inverse Laplace transform for convection-diffusion equations
For researchers solving convection-diffusion equations, this method offers a general algorithmic approach to contour selection without a priori pseudospectral information, but the improvement is incremental over existing contour methods.
The paper proposes a novel contour integral method for numerically inverting the Laplace transform to solve linear convection-diffusion equations, using an elliptic arc contour and algorithmic pseudospectral level set selection. Numerical experiments on Black-Scholes (1D) and Heston (2D) equations show the method is competitive with existing contour integral methods.
In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is competitive with other contour integral methods available in the literature.