Componentwise Splitting Method Research Articles

An implicit direct unconditionally stable numerical algorithm for the solution of advection-diffusion equation on a sphere

A new algorithm, proposed for solving linear and nonlinear advection-diffusion problems on a sphere, is tested with various numerical experiments. The velocity field on the sphere is non-divergent, and assumed to be known. Discretization of the advection-diffusion equation in space is performed by the finite volume method using the Gauss theorem for each grid cell. For the discretization in time, the symmetrized double-cycle componentwise splitting method and the Crank-Nicolson scheme are used. The one-dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman-Morrison's formula and Thomas's algorithm. The one-dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The algorithm is of second order approximation in space and time. It is implicit, unconditionally stable, direct (without iterations) and rapid in realization. The theoretical results are confirmed numerically by simulating various linear and nonlinear advection-diffusion processes. The tests show high accuracy and efficiency of the method that correctly describes the advection-diffusion processes and the mass balance of a substance in a forced and dissipative discrete system. In addition, in the absence of external forcing and dissipation, it conserves both the total mass and the norm of solution.

Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility, stochastic interest rates and stochastic mortality via the componentwise splitting method

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

SummaryA numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L2‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.

Casting Filling Simulation Technology Based on Component Wise Splitting Method

The casting filling numerical simulation technology is of great significance to optimize the casting process, improve castings quality, and reduce production costs. Component wise splitting method only uses a small amount of iterations between velocity field and pressure field, so it has high computational efficiency in theory. The calculation accuracy is analyzed through Lid-driven cavity flow model, the result has a good agreement with literature results in low Reynolds number, but there is a big deviation when high Reynolds number. Finally the fraction step method is applied to the solution of the low pressure casting filling process, Calculation results coincided with the experimental results.

THE EVALUATION OF AMERICAN OPTION PRICES UNDER STOCHASTIC VOLATILITY AND JUMP-DIFFUSION DYNAMICS USING THE METHOD OF LINES

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

Efficient numerical methods for pricing American options under stochastic volatility

AbstractFive numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY

Efficient numerical methods for pricing American options using Heston's stochastic volatility model are proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a linear complementarity problem. The idea of operator splitting methods is to divide each time step into fractional time steps with simpler operators. This paper proposes componentwise splitting methods for solving the linear complementarity problem. The basic componentwise splitting decomposes the discretized problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz algorithm, which was originally introduced for American options under the Black and Scholes model. The accuracy of the componentwise splitting method is increased by applying the Strang symmetrization. The good accuracy and the computational efficiency of the proposed symmetrized splitting method are demonstrated by numerical experiments.

Computer simulation of the solidification of cast titanium dental prostheses

Temperature distributions in titanium dental castings and molds are of great influence on the quality of titanium dental castings, and few efforts have been made in the numerical simulation of heat transfer in the process for casting titanium for dental applications. A finite difference scheme of the component-wise splitting method, which is unconditionally stable, was developed to solve the three-dimensional heat transfer problem for titanium dental casting during the investment cast and centrifugal cast process. 4 kinds of runner system were simulated and the computational efficiency were analyzed by the component-wise splitting method and the explicit finite difference method, the results shown that the techniques used in the current research can greatly improve the computational efficiency of the simulation system. The porosity predictions of 4 kinds of runner system were carried out with the simulation program. The predicted results were in good agreement with those of literatures.

Numerical methods to improve the computational efficiency of thermal analysis for the die casting process

The temperature distributions in the die casting and the dies are of great influence on the quality of the die casting and the life cycle of the dies, so that many efforts have been made in the numerical simulation of heat transfer in the die casting process. In order to improve the computational efficiency of thermal analysis of the die casting process, three kinds of numerical methods were employed. First, the component-wise splitting method was adopted to solve the three-dimensional heat transfer problem in the die casting process. The method is unconditionally stable and suitable for simulating fast transient phenomena and for computations on fine meshes. Secondly, an irregular mesh system was developed to reduce the total grid number of the analysis system. The corresponding finite difference algorithm of the component-wise splitting method was also developed. Thirdly, a transient surface layer concept was used to divide the computational area into a transient area and a steady area, where different time steps can be applied to these areas to improve the computational efficiency. A three-dimensional thermal analysis system was developed with the numerical methods and a practical die casting component was simulated with focus on the computational efficiency.

An efficient thermal analysis system for the die-casting process

An absolutely stable finite difference scheme, the component-wise splitting method, has been introduced and adopted in the mathematical modeling of the heat-transfer problem in the die-casting process. Technical considerations related to the mathematical modeling processes, such as the boundary conditions and the irregular enmeshment scheme, etc., are also presented. Based on the mathematical modeling described above, a 3D thermal analysis system for the die-casting process has been developed. The verification and application examples show that the system can greatly improve the computational efficiency in two ways: (i) the irregular mesh size can greatly reduce the grid number of the analysis system; and (ii) a relatively larger time step can be used in the simulation process since the component-wise splitting method is absolutely stable.

A generalized two‐cycle componentwise splitting method for solving three‐dimensional parabolic differential equations with variable coefficients on multilayers

Describes a generalized two‐cycle componentwise splitting method for solving three‐dimensional parabolic differential equations with variable coefficients which has been developed based on the idea of the regularized difference scheme. The method is simple, unconditionally stable and well suited for simulating fast transient phenomena and for computations on fine spatial meshes. A numerical procedure that employs the generalized two‐cycle componentwise splitting scheme was developed to solve three‐dimensional parabolic differential equations with variable coefficients on multilayers. In the procedure, the generalized “divide and conquer” method for solving tridiagonal linear systems is applied in order to overcome the problem with the unknown value at the interface between layers. Numerical results show that the procedure is accurate and efficient.