Supraconvergent cell-centered scheme for two dimensional elliptic problems

Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique

A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

Mesh stretch effects on convection in flow simulations

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

In this study, the representation of discretization error using Taylor series in finite difference solutions is investigated as well as the behavior of the exact solutions to the finite difference equations as a function of the grid size and grid refinement factor. The results are compared to the classical Richardson Extrapolation method whereby the numerical solution (or the error) is explicitly expressed as a Taylor series expansion. The exact finite difference solutions are used to demonstrate that oscillatory convergence is a common occurrence. The expansion of the numerical solutions in Taylor series is based on the exact finite difference solutions that are obtained using different discretization schemes. It is shown that in some cases the numerical solution exhibited a singular behavior which can not be remedied easily. Some exact finite difference solutions also exhibited oscillatory behavior which was not due to the use of mixed order terms as is usually believed by the Computational Fluid Dynamics community. Moreover, representation of the numerical solution using Taylor series is not always satisfactory even in case of relatively simple one-dimensional problems.

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O ( Δ x 3 - α ) ${O(\Delta x^{3- \alpha })}$ , 1 < α < 2 ${1<\alpha <2}$ . A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O ( Δ t + Δ x min ( 3 - α , β ) ) ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$ , 1 < α < 2 ${1< \alpha <2}$ , β > 0 ${\beta >0}$ , where Δ t , Δ x ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

We present the mathematical equations that govern heat transfer in a polymer melt flowing in a circular tube with constant ambient temperature, taking into account the viscous dissipation effects. This leads to a nonlinear parabolic partial differential equation. It is shown that the exact solution of a linearized version of the governing equation can be presented in terms of the Whittaker function. A finite difference scheme is used to produce an approximate solution of the linearized problem. This numerical solution is shown to be a good approximation to the exact solution found in terms of the Whittaker function. The finite difference scheme is then modified to approximate the nonlinear parabolic partial differential equation and is compared with the results found using the finite element method. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 289–294, 2006

The foundation has a function to hold the load in the form of axial load, lateral load and moment. The axial load on the pile foundation is supported by the pole end resistance and pile friction, while the lateral load on the pile foundation is supported by the relation of the blanket area along with the pile with the ground in the lateral direction. Lateral load causes lateral deflection in which the magnitude of deflection must not exceed the permissible lateral deflection limit of 2.54 cm. The determining factors in lateral deflection are the type of pile foundation, soil type, and the magnitude of the force that occurs. The foundation of a single pile in which the number of one pile, when embedded in clay soil, needs to be investigated for the magnitude of lateral deflection that occurs, this is due to the changing nature of the clay when under the influence of high or low water content. The foundation of the pile uses a concrete head with a condition of a free end with a cylindrical shape with a diameter of 60 cm that is not solid with a wall thickness of 10 cm and a compressive strength of concrete of 60 MPa. Clay soil data uses soil drilling test data in Wirosari, Grobogan, Central Java. The planning method used is the finite element method (PLAXIS program) and the finite difference method (py curve and the ALLPILE program) with lateral load variations of 10 kN, 15 kN and 20 kN given to the top end of the pole with the principle of giving trial and error loads. The results of the calculation of the lateral deflection of the pile on the py curve method with a finite-difference resolution, the ALLPILE program, and the PLAXIS program respectively with a lateral load of 10 kN are 0.0629 cm; 1.21 cm; 0.27 cm, lateral loads of 15 kN are 0.0943 cm; 2.13 cm; 0.4051 cm, and a lateral load of 20 kN is 0.1258 cm; 3.14 cm; 0.5402 cm. Thus the lateral deflection load limit is 15 kN, so as not to exceed the permissible lateral deflection limit of 2.54 cm. The recommended method used is the ALLPILE program to better get the level of security from the lateral deflection of the pole when applied in the field

Finite element and finite difference solutions are obtained for transient temperature distribution in a partially cooled cylindrical rod that generates heat at a uniform rate. A portion of the rod is immersed in a coolant reservoir that is maintained at constant temperature, and the exposed portion of the rod is cooled by convective heat transfer. Because thermal conductivity of the rod is temperature dependent, the governing partial differential equation is nonlinear. The analytical techniques utilized in solving the problem could be applied to analyzing the cooling of spent nuclear fuel rods. The finite difference method used to solve the problem utilizes an implicit formulation of the governing equation, and a numerical technique for handling the nonlinear terms. Validation of the numerical solution is obtained by comparing the results at a specified time against those generated by a commercial finite element software package. The computer model for the problem was used to estimate heat generation rates that could initiate meltdown of a fuel rod.

Analytical Measurement

Pricing TARN Using a Finite Difference Method