IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916377501106176 |
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| author | López-Salas, J. G. Suárez-Taboada, M. Castro, M. J. Ferreiro-Ferreiro, A. M. García-Rodríguez, J. A. |
| author_facet | López-Salas, J. G. Suárez-Taboada, M. Castro, M. J. Ferreiro-Ferreiro, A. M. García-Rodríguez, J. A. |
| contents | The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing López-Salas, J. G. Suárez-Taboada, M. Castro, M. J. Ferreiro-Ferreiro, A. M. García-Rodríguez, J. A. Numerical Analysis Analysis of PDEs The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order. |
| title | IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2409.01125 |