In this study, we solve the Fokker-Planck equation by a COMPACT finite difference METHOD, By the finite difference METHOD the computation of Fokker-Planck equation is reduced to a system of ordinary differential equations. Two different METHODs, boundary value METHOD and cubic C1-spline collocation METHOD, for solving the resulting system are proposed. Both METHODs have fourth-order accuracy in time variable. By the boundary value METHOD, some pointwise approximate solutions are only obtained. But, C1-spline METHOD gives a closed-form approximation in each space step, too. Illustrative examples are included to demonstrate the validity and efficiency of the METHODs. A comparison is made with existing results.

This paper is devoted to applying the sixth-order COMPACT finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the COMPACT finite difference METHOD. Through this way, the computational costs of the METHOD with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.

This paper aims to apply and investigate the COMPACT finite difference METHODs for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of COMPACT finite difference. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the convergence analysis of the proposed approach for the fractional and nonfractional cases and prove that the METHODs are convergent under some suitable conditions. Examples are also given to illustrate the efficiency of our METHOD compared to other METHODs.

Mode I fracture toughness (KIC) is one of the most important parameters in the fracture mechanics of brittle material. Several laboratory METHODs have been suggested to determine the mode I fracture toughness. However, many of these METHODs deal with the lengthy sample preparation procedure, premature failure of samples, and difficulties in obtaining the precise value of the fracture toughness property. In this paper, a recently proposed pseudo-COMPACT tension METHOD is used to evaluate mode I fracture toughness of a middle-grain granite benefiting the advantages of this METHOD including; simplicity of the test, high level of test control, and high accuracy of the KIC value. For this purpose, granite samples in four different diameters and with six test repeats per diameter have been prepared and tested using the pseudo-COMPACT tension METHOD. For each sample, in addition to recording the load and displacement data, the acoustic events during the loading process were also recorded simultaneously by an acoustic emission equipment. First, the resulting fracture toughness value for each sample has been determined, then the size effect has been evaluated and analyzed. Finally, the results of the acoustic emission METHOD, as the monitoring tool in the fracturing process of tested samples, have been analyzed. The qualitative evolution of acoustic emission parameters well illustrates the mechanical process occurring in the tested samples with well-matched coinciding with the mechanical transitions observed in samples during the loading process. Experimental results show that mode I fracture toughness is positively related to the specimen size and there is a noticeable size effect in KIC value up to a certain diameter.

The dimensionless form of Navier-Stokes equations for two dimensional jet flows are solved using direct numerical simulation. The length scale and the velocity scale of jet flow at the inlet boundary of computational domain are used as two characteristics to define the jet Reynolds number. These two characteristics are jet half-width and centerline velocity. Governing equations are discretized in streamwise and cross stream directions using a sixth order COMPACT finite difference scheme and a mapped COMPACT finite difference METHOD, respectively. Cotangent mapping of y=-b cot (pz) is used to relate the physical domain of y to the computational domain of z. The COMPACT third order Runge-Kutta METHOD is used for time-advancement of the simulation. convective outflow boundary condition is employed to create a non-reflective type boundary condition at the outlet. An inviscid Stuart flow and a completely viscose solution of Navier Stokes equations are used for the verification of numerical simulations. Results for perturbed jet flow in self-similar coordinates were also investigated which indicate that the time-averaged statistics for velocity, vorticity, turbulence intensities and Reynolds stress distribution tend to collapse on top of each other at flow downstream locations.

Usually, simplified models, such as shallow water model, are used to describe atmospheric and oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. This model is applied to a fluid layer of constant density in which the horizontal scale of the flow is much greater than the layer depth. However, the dynamics of a two-dimensional shallow water model is less general than three-dimensional general circulation models but is preferred because of its greater mathematical and computational simplicity.Taking intrinsic complexity of fluids, recently, numerical researches have been focused on highly accurate METHODs. Especially, for large grid spacing numerical simulation, the use of highly accurate METHODs have become urgent. This trend led to an interest in COMPACT finite difference METHODs. The COMPACT finite-difference schemes are simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, COMPACT schemes have proved to be significantly more accurate along with the benefits of using smaller stencil sizes, which can be essential in treating non-periodic boundary conditions. Application of some families of the COMPACT schemes to the spatial differencing in some idealized models of the atmosphere and oceans shows that COMPACT finite difference schemes can be considered as a promising METHOD for the numerical simulation of geophysical fluid dynamics problems.In this research work, the sixth-order combined COMPACT (CCD6) finite difference METHOD was applied to the spatial differencing of f-plane shallow-water equations in vorticity, divergence and height forms (on a Randall's Z grid). The second-order centered (E2S), fourth-order COMPACT (C4S) and sixth-order super COMPACT (SCD6) finite difference METHODs were also used for spatial differencing of the shallow water equations and the results were compared to the ones from a pseudo-spectral (PS) METHOD. A perturbed unstable zonal jet was considered as the initial condition for numerical simulation in which it breaks up into smaller vortices and becomes very complex. The shallow water equations are integrated in time using a three-level semi-implicit formulation. To control the build-up of small-scale activities and thus potential for numerical nonlinear instability, the non-dissipative vorticity equation was made dissipative by adding a hyperdiffusion term. The global distribution of mass between isolevels of the potential vorticity, called mass error, was used to assess numerical accuracy. The CCD6 generated the least mass error among finite difference METHODs used in this research. By taking the PS METHOD as a reference, the qualitative and quantitative comparison of the results of the CCD6, SCD6, C4S and E2S, indicated the high accuracy of the sixth-order combined COMPACT finite difference METHOD.