Finite analytic numerical method for unsteady two-dimensional Navier-Stokes equations

Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.

Finite-Analytic Numerical Solution of Heat Transfer in Two-Dimensional Cavity Flow

The unsaturated zone plays an important role in groundwater recharge, discharge, and the ecological environment. Mathematical models are essential to the investigations of flow processes in the unsaturated zone. The governing equations of the mathematic models for unsaturated flow are nonlinear since their unsaturated hydraulic parameters of the equations depend on their solutions (i.e., pressure or moisture content). The nonlinearity can become very strong with flow in relatively dry soils, and the equation has hyperbolic characteristics. Therefore, the traditional numerical methods, such as the finite difference method and the finite element method, could lead to the numerical oscillation, dispersion, and the divergence of the solution if improper time and space steps were selected. In this study, a finite analytic method (FAM) to solve Richards’ equation was developed. The basic idea of the FAM is the incorporation of the analytic solution in a small local element to formulate the algebraic representation of the partial differential equation of the unsaturated flow. Therefore, the finite analytic numerical method can effectively control the numerical oscillation and dispersion. The convergence and stability of finite analytic numerical scheme are then proven by a rigorous mathematical analysis. Numerical experiments are then used to show that FAM is highly accurate by comparing its results with those obtained using analytical solutions by a modified Picard finite difference method. In addition, we also demonstrate that FAM reproduces results of laboratory experiments. Therefore, it can be considered an appropriate simulation method for the unsaturated flow.

Finite Analytic Method for Solving the Unsaturated Flow Equation

A hybrid Laplace transform finite analytic method for solving transport problems with large Peclet and Courant numbers

Two methods for developing high order single step explicit algorithms on symmetric stencils with data on only one time level are presented. Examples are given for the convection and linearized Euler equations with up to the eighth order accuracy in both space and time in one space dimension, and up to the sixth in two space dimensions. The method of characteristics is generalized to nondiagonalizable hyperbolic systems by using exact local polynominal solutions of the system, and the resulting exact propagator methods automatically incorporate the correct multidimensional wave propagation dynamics. Multivariate Taylor or Cauchy-Kowaleskaya expansions are also used to develop algorithms. Both of these methods can be applied to obtain algorithms of arbitrarily high order for hyperbolic systems in multiple space dimensions. Cross derivatives are included in the local approximations used to develop the algorithms in this paper in order to obtain high order accuracy, and improved isotropy and stability. Efficiency in meeting global error bounds is an important criterion for evaluating algorithms, and the higher order algorithms are shown to be up to several orders of magnitude more efficient even though they are more complex. Stable high order boundary conditions for the linearized Euler equations are developed in one space dimension, and demonstrated in two space dimensions.

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An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case

A new numerical method called the Finite Analytical Method for solving partial differential equations is introduced. The basic idea of the finite analytic method is the incorporation of the local analytic solution in obtaining the numerical solution of the problem. The finite analytical method first divides the total region of the problem into small subregions in which local analytic solutions are obtained. Then an algebraic equation is derived from the local analytic solution for each subregion relating an interior nodal value at a point P in the subregion to its neighboring nodal values. The assembly of all the local analytic solutions thus provides the finite-analytic numerical solution of the problem. In this paper the finite analytic method is illustrated in solving steady and unsteady heat transfer problems.

The Finite Analytic Method for Steady and Unsteady Heat Transfer Problems

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The Finite Analytic Method for Steady and Unsteady Heat Transfer Problems