Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation

2012 ◽  
Vol 16 (1) ◽  
pp. 83-110 ◽  
Author(s):  
Tinne Haentjens ◽  
Karel In 't Hout
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Partial Differential ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Implicit Finite Difference

scholarly journals An Efficient Alternating Direction Explicit Method for Solving a Nonlinear Partial Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Somayeh Pourghanbar ◽  
Jalil Manafian ◽  
Mojtaba Ranjbar ◽  
Aynura Aliyeva ◽  
Yusif S. Gasimov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Nonlinear Partial Differential Equation ◽  
Finite Difference Schemes ◽  
Explicit Method ◽  
Partial Differential ◽  
Unconditionally Stable ◽  
Elapsed Time ◽  
Alternating Direction

In this paper, the Saul’yev finite difference scheme for a fully nonlinear partial differential equation with initial and boundary conditions is analyzed. The main advantage of this scheme is that it is unconditionally stable and explicit. Consistency and monotonicity of the scheme are discussed. Several finite difference schemes are used to compare the Saul’yev scheme with them. Numerical illustrations are given to demonstrate the efficiency and robustness of the scheme. In each case, it is found that the elapsed time for the Saul’yev scheme is shortest, and the solution by the Saul’yev scheme is nearest to the Crank–Nicolson method.

scholarly journals Higher-Order Linear-Time Unconditionally Stable Alternating Direction Implicit Methods for Nonlinear Convection-Diffusion Partial Differential Equation Systems

2014 ◽  
Vol 136 (6) ◽  
Author(s):  
Oscar P. Bruno ◽  
Edwin Jimenez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Time ◽  
High Order Accuracy ◽  
Implicit Methods ◽  
Partial Differential ◽  
Nonlinear Convection ◽  
Alternating Direction Implicit ◽  
Convection Diffusion ◽  
Alternating Direction

We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDFs) of order p≥2, for the numerical solution of two-dimensional, time-dependent, nonlinear, convection-diffusion partial differential equation (PDE) systems in Cartesian domains. The proposed algorithms, which do not require the solution of nonlinear systems, additionally produce solutions of spectral accuracy in space through the use of Chebyshev approximations. In particular, these methods give rise to minimal artificial dispersion and diffusion and they therefore enable use of relatively coarse discretizations to meet a prescribed error tolerance for a given problem. A variety of numerical results presented in this text demonstrate high-order accuracy and, for the particular cases of p=2,3, unconditional stability.

scholarly journals New schemes for a two-dimensional inverse problem with temperature overspecification

2001 ◽  
Vol 7 (3) ◽  
pp. 283-297 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Central Processor ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Alternating Direction

Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

A Simple and Stable Numerical Solution for the Population Density Equation

2003 ◽  
Vol 15 (9) ◽  
pp. 2129-2146 ◽  
Author(s):  
M. de Kamps
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Population Density ◽  
Finite Difference ◽  
Method Of Characteristics ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Integrate And Fire ◽  
Large Populations ◽  
Density Equation

A population density description of large populations of neurons has generated considerable interest recently. The evolution in time of the population density is determined by a partial differential equation (PDE). Most of the algorithms proposed to solve this PDE have used finite difference schemes. Here, I use the method of characteristics to reduce the PDE to a set of ordinary differential equations, which are easy to solve. The method is applied to leaky-integrate-and-fire neurons and produces an algorithm that is efficient and yields a stable and manifestly nonnegative density. Contrary to algorithms based directly on finite difference schemes, this algorithm is insensitive to large density gradients, which may occur during evolution of the density.

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