Finite Volume Methods

A second order finite volume IMEX Runge-Kutta scheme for two dimensional PDEs in finance ArXiv ID: 2410.02925 “View on arXiv” Authors: Unknown Abstract In this article we present a novel and general methodology for building second order finite volume implicit-explicit (IMEX) numerical schemes for solving two dimensional financial parabolic PDEs with mixed derivatives. In particular, applications to basket and Heston models are presented. The obtained numerical schemes have excellent properties and are able to overcome the well-documented difficulties related with numerical approximations in the financial literature. The methods achieve true second order convergence with non-regular initial conditions. Besides, the IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing very accurate and non-oscillatory approximations of the Greeks. Finally, in order to assess all the aforementioned good properties of the developed numerical schemes, we compute extremely accurate semi-analytic solutions using multi-dimensional Fourier cosine expansions. A novel technique to truncate the Fourier series for basket options is presented and it is efficiently implemented using multi-GPUs. ...

Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs ArXiv ID: 2410.02927 “View on arXiv” Authors: Unknown Abstract In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes. ...