Due to improvements in computational resources, interest has recently increased in using implicit scheme for solving flow equations on 3D unstructured grids. However, most of the implicit schemes produce greater numerical diffusion error than their corresponding explicit schemes. This stems from the fact that in linearizing implicit fluxes, it is conventional to replace the Jacobian matrix in t...

Adaptive solution techniques are presented for simulating underwater explosions and implosions. The liquid is assumed to be an adiabatic fluid and the solution in the gas is assumed to be uniform in space. The solution in the water is integrated in time using a semi-implicit time discretization of the adiabatic Euler equations. Results are presented either using a non-conservative semi-implicit...

Many temporal discretization methods for linear evolution equations converge uniformly on compact time intervals at the rate 1 nα only for sufficiently smooth initial data. It is shown that these methods can be regularized such that the new schemes converge ‘in the average’ at the rate 1 nα for all initial data. Examples given include the Crank-Nicholson scheme and the alternating direction imp...

In the present investigation, new explicit approaches by Milstein method and increment function of Jacobian derivative drift coefficient are designed. Several numerical tests such as Cox–Ingersoll–Ross process, stochastic Brusselator, Davis-Skodje system presented to illustrate accuracy efficiency our schemes. Furthermore, we show that strong convergence rate procedures is approximately one.

The (Extrapolated) Alternating Direction Implicit Preconditioners for the class of Conjugate Gradient Methods are applied for the solution of the second order elliptic equation in a rectangle under Dirichlet boundary conditions. The PDE is approximated by uniform meshes of 5− and 9−point difference schemes and analytic expressions for the optimal acceleration and extrapolation parameters are ob...

This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability prope...

Maximum drawdown is a risk measure that plays an important role in portfolio management. In this paper, we address the question of computing the expected value of the maximum drawdown using a partial differential equation (PDE) approach. First, we derive a two-dimensional convection-diffusion pricing equation for the maximum drawdown in the Black-Scholes framework. Due to the properties of the ...

Due to improvements in computational resources, interest has recently increased in using implicit scheme for solving flow equations on 3D unstructured grids. However, most of the implicit schemes produce greater numerical diffusion error than their corresponding explicit schemes. This stems from the fact that in linearizing implicit fluxes, it is conventional to replace the Jacobian matrix in t...