AMF-type W-methods for Parabolic Problems with Mixed Derivatives

González-Pinto, S.; Hairer, Ernst; Hernández‐Abreu, D.; Pérez-Rodríguez, S.

doi:10.1137/17m1163050

Cited by 16 publications

(6 citation statements)

References 27 publications

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“…The first goal of this section is to show unconditional convergence of order two in the maximum norm for semilinear parabolic problems with constant diffusion coefficients (and a time dependent source term) and time-independent Dirichlet boundary conditions ( 1)-( 2), when the one-step AMF-W-method (henceforth denoted as AMF-W1) in [4,6] is considered with the parameter choice θ = 1/2…”

Section: Convergence In the Uniform Norm Of Some Adi-type Methodsmentioning

confidence: 99%

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Convergence in the maximum norm of ADI-type methods for parabolic problems

González-Pinto

Hernández‐Abreu

2022

Applied Numerical Mathematics

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Section: Convergence In the Uniform Norm Of Some Adi-type Methodsmentioning

confidence: 99%

“…. , z m ) is a rational function that satisfies (6), then there exists a constant K only depending on α and m such that (5) holds for C = K/ β, β =: min 1≤j≤m β j > 0.…”

Section: Bounds In the Maximum Norm For Rational Functionsmentioning

confidence: 99%

Convergence in the maximum norm of ADI-type methods for parabolic problems

González-Pinto

Hernández‐Abreu

2022

Applied Numerical Mathematics

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“…In this work our focus is on the time integration of the differential systems that come from some spatial discretization of (1)-( 2)-(3) by using Finite Differences. In particular, we are interested in AMF-type W-methods (based on Approximate Matrix Factorization) which avoid the solution of non-linear equations and only require the solution of linear systems with tridiagonal matrices, as it can be seen in [7], [6], [5]. They have also been successful in applications to the case of variable coefficients in [9] and multi-dimensional problems (N > 3) in [13].…”

Section: Introductionmentioning

confidence: 99%

Boundary corrections on multi-dimensional PDEs

González-Pinto

Hernández‐Abreu

Pérez-Rodríguez

2023

Preprint

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Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated to the numerical solution of initial boundary value problems for parabolic Partial Differential Equations in arbitrary spatial dimensions with time dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similarly as possible as for the interior points of the domain. These new techniques are considered for the time integration with W-methods based on Approximate Matrix Factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

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“…For multidimensional option valuation PDEs (without integral part), operator splitting schemes are widely considered in the literature. Notably, alternating direction implicit (ADI) methods, which combine implicit unidirectional correction stages with explicit stages involving the mixed derivative terms, are highly effective, see for example [16,21,23,25,40]. Hundsdorfer & in 't Hout [18] recently studied a novel class of multistep stabilizing correction methods for PDEs with mixed derivative terms and showed that such methods can be competitive with the well-established onestep modified Craig-Sneyd (MCS) scheme [25] in the application to the familiar two-dimensional Heston PDE [17].…”

Section: Introductionmentioning

confidence: 99%

Operator splitting schemes for the two-asset Merton jump–diffusion model

Boen

Hout

2021

Journal of Computational and Applied Mathematics

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This paper deals with the numerical solution of the two-dimensional time-dependent Merton partial integro-differential equation (PIDE) for the values of rainbow options under the two-asset Merton jump-diffusion model. Key features of this well-known equation are a twodimensional nonlocal integral part and a mixed spatial derivative term. For its efficient and stable numerical solution, we study seven recent and novel operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. Here the integral part is always conveniently treated in an explicit fashion. The convergence behaviour and the relative performance of the seven schemes are investigated in ample numerical experiments for both European put-on-the-min and put-on-the-average options.

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AMF-type W-methods for Parabolic Problems with Mixed Derivatives

Cited by 16 publications

References 27 publications

Convergence in the maximum norm of ADI-type methods for parabolic problems

Convergence in the maximum norm of ADI-type methods for parabolic problems

Boundary corrections on multi-dimensional PDEs

Operator splitting schemes for the two-asset Merton jump–diffusion model

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