Analysis of a stabilized finite element approximation of the transient convection-diffusion-reaction equation using orthogonal subscales

In this paper we analyze a stabilized finite element method to solve the transient convection-diffusion-reaction equation based on the decomposition of the unknowns into resolvable and subgrid scales. We start from the time-discrete form of the problem and obtain an evolution equation for both components of the decomposition. A closed-form expression is proposed for the subscales which, when inserted into the equation for the resolvable scale, leads to the stabilized formulation that we analyze. Optimal error estimates in space are provided for the first order, backward Euler time integration. Received: 31 January 2001 / Accepted: 30 September 2001     

A new treatment of capillarity to improve the stability of IMPES two-phase flow formulation

In this paper, we present an efficient numerical method for two-phase immiscible flow in porous media with different capillarity pressures. In highly heterogeneous permeable media, the saturation is discontinuous due to different capillary pressure functions. One popular scheme is to split the system into a pressure and a saturation equation, and to apply IMplicit Pressure Explicit Saturation (IMPES) approach for time stepping. One disadvantage of IMPES is instability resulting from the explicit treatment for capillary pressure. To improve stability, the capillary pressure is usually incorporated in the saturation equation which gradients of saturation appear. This approach, however, does not apply to the case of different capillary pressure functions for multiple rock-types, because of the discontinuity of saturation across rock interfaces. In this paper, we present a new treatment of capillary pressure, which appears implicitly in the pressure equation. Using an approximation of capillary function, we substitute the implicit saturation equation into the pressure equation. The coupled pressure equation will be solved implicitly and followed by the explicit saturation equation. Five numerical examples are provided to demonstrate the advantages of our approach. Comparison shows that our proposed method is more efficient and stable than the classical IMPES approach.     

Risk sensitive control of Markov processes in countable state space

In this paper we consider infinite horizon risk-sensitive control of Markov processes with discrete time and denumerable state space. This problem is solved by proving, under suitable conditions, that there exists a bounded solution to the dynamic programming equation. The dynamic programming equation is transformed into an Isaacs equation for a stochastic game, and the vanishing discount method is used to study its solution. In addition, we prove that the existence conditions are also necessary.     

On the small time behavior of the nonlinear estimation problem for finite bandwidth signals

In this paper we investigate the small time behavior of solutions of the Zakai equation. We derive a wave equation-like stochastic partial differential equation which is related to the Zakai equation. We are able to solve this equation for sufficiently smooth signals, and (approximately) transform these into solutions of the Zakai equation. We construct a Hadamardtype expansion for solutions of this partial differential equation and show how this expansion is related to a small time expansion of solutions of the Zakai equation.     

A note to the integrable discretization of the mKdV and Schrödinger equations

An approach has been proposed to the integrable discretization of nonlinear evolution equations. Based on the bilinear formalism, we choose appropriate substitution from hyperbolic operator into continuous Hirota operators and obtain several new kinds of integrable system through seeking their 3-soliton solutions, such as the mKdV equation, the nonlinear Schrödinger equation and so on. By applying Adomian decompose method, we discuss the numerical analysis property to the discrete mKdV equation. In addition, we also point out the relations between the above discreted equations and some well-known equations.     

A Fitted Finite Volume Method for the Valuation of Options on Assets with Stochastic Volatilities

In this paper, we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presented.     

A Neural Network Study of Blasius Equation

In this work we applied a feed forward neural network to solve Blasius equation which is a third-order nonlinear differential equation. Blasius equation is a kind of boundary layer flow. We solved Blasius equation without reducing it into a system of first order equation. Numerical results are presented and a comparison according to some studies is made in the form of their results. Obtained results are found to be in good agreement with the given studies.

     

Fast and efficient numerical method for solving the Allen–Cahn equation on the cubic surface

In this study, we present a fast and efficient finite difference method (FDM) for solving the Allen–Cahn (AC) equation on the cubic surface. The proposed method applies appropriate boundary conditions in the two-dimensional (2D) space to calculate numerical solutions on cubic surfaces, which is relatively simpler than a direct computation in the three-dimensional (3D) space. To numerically solve the AC equation on the cubic surface, we first unfold the cubic surface domain in the 3D space into the 2D space, and then apply the FDM on the six planar sub-domains with appropriate boundary conditions. The proposed method solves the AC equation using an operator splitting method that splits the AC equation into the linear and nonlinear terms. To demonstrate that the proposed algorithm satisfies the properties of the AC equation on the cubic surface, we perform the numerical experiments such as convergence test, total energy decrease, and maximum principle.     

一类强非线性系统共振周期解的渐近分析

强非线性系统经引入参数变换,并在一定的假设条件下,可转化为弱非线性系统．将其解展成为改进的傅立叶级数后,利用参数待定法可方便地求出强非线性系统的共振周期解．研究了Duffing方程的主共振、Van der Pol方程的3次超谐共振和Van der Pol-Mathieu方程的1／2亚谐共振周期解．这些例子表明近似解与数值解非常吻合。     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.