Results For Differential Equations Research Articles

A New Numerical Scheme for Convection-Dominated Transport Equations.

In order to obtain numerical solutions stable and accurate for convection-dominated transport equations, we propose a criterion in constructing numerical schemes for the convection term that roots of the characteristic equation for resulting difference equation have poles. By imposing this criterion on the difference coefficients for the convection term, we construct a new numerical scheme robust for convection-dominated equations. The present new scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is 8/3, which is the critical value for its stability, while it approaches the second-order upwind scheme as Rm goes to infinity. Hence the present scheme interpolates a stable scheme between the QUICK scheme at Rm=8/3 and the second-order upwind scheme at Rm=infinity. This new scheme shows good numerical solutions for one-dimensional, convection-diffusion equations.

New numerical schemes based on a criterion for constructing essentially stable and accurate numerical schemes for convection‐dominated equations

AbstractIn order to obtain stable and accurate numerical solutions for the convection‐dominated steady transport equations, we propose a criterion for constructing numerical schemes for the convection term that the roots of the characteristic equation of the resulting difference equation have poles.By imposing this criterion on the difference coefficients of the convection term, we construct two numerical schemes for the convection‐dominated equations. One is based on polynomial differencing and the other on locally exact differencing.The former scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is \documentclass{article}\pagestyle{empty}\begin{document}$\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $\end{document}, which is the critical value for its stability, while it approaches the second‐order upwind scheme as Rm goes to infinity. Hence the former scheme interpolates a stable scheme between the QUICK scheme at Rm = \documentclass{article}\pagestyle{empty}\begin{document}$\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $\end{document} and the second‐order upwind scheme at Rm = ∞. Numerical solutions with the present new schemes for the one‐dimensional, linear, steady convection‐diffusion equations showed good results.

A full finite difference time domain implementation for radio wave propagation in a plasma

A full finite difference time domain methodology is developed for electromagnetic wave propagation in a plasma. The finite difference grid is consistent with central difference approximation of the curl, divergence and gradient operators that appear in the joint equations of Euler and Maxwell, and the coupling effects between the fluid velocity and the electric field. To accomplish the time advancement, the central difference approximation is invoked for the time derivatives and leapfrog concepts are employed. The resulting difference equations converge to the exact equations, provided that the developed stability requirement is satisfied. Finally, numerical results are provided and compared with the inverse fast Fourier transform results of closed‐form, frequency domain solutions for the half space problem; the agreement between solutions is shown to be excellent.

Discretization and truncation errors in a numerical solution of Laplace’s equation

Numerical solutions to Laplace’s equation for an electrostatic potential can easily be found in undergraduate physics courses by approximating the Laplacian on a mesh and solving the resulting difference equations using a spreadsheet program or a simple program written in basic or pascal. The simplest numerical method uses iteration and accelerates the convergence by simultaneous overrelaxation (SOR). Truncating the iteration introduces an error in the solution to the difference equations, and this raises the question of how stringent to make the criterion for convergence. This paper considers the relative magnitude of the errors made in approximating the Laplacian (discretization error) and in truncating the iteration (truncation error). Numerical results are given for an electrostatic cavity problem previously investigated by several authors, and the numerical solutions are compared with an exact solution obtained by conformal mapping. It was found that when even a modest convergence criterion is used to truncate the iteration, the rms error inherent in discretization is more than an order of magnitude larger than the error in the solution of the difference equations. It was also found that the commonly used nearest-neighbor approximation to the Laplacian gives the most accurate numerical solutions.

COVOLUME-DUAL VARIABLE METHOD FOR THERMALLY EXPANDABLE FLOW ON UNSTRUCTURED TRIANGULAR GRIDS

SUMMARY Finite difference like discretizations are developed for the time dependent Navier-Stokes equations and the thermal energy equation. The flow is assumed to be thermally expandable, that is, the density varies only with temperature. A new pointwise first order upwind scheme for convection is presented which is of nonnegative type. Also presented are new approaches to reconstructing the velocity vector field from the covolume primitive variables. The resulting difference equations reproduce linear flow fields.

Multigrid Solution to Steady-State Two-Dimensional Conservation Laws

A new genuinely two-dimensional compact approach to discretizing scalar steady-state conservation laws is presented. It provides the possibility to separate treatment of the streamwise and cross-stream directions. Due to this separation, the artificial viscosity can be added in the streamwise direction only. A high-resolution mechanism is introduced in the cross-stream direction. The resulting schemes are shown to be second-order accurate and monotonic, providing a good resolution of discontinuities, representing them in the numerical solution by this oscillation-free transition layers. Due to their good stability properties, the resulting difference equations can be solved efficiently by a multigrid algorithm employing a simple pointwise relaxation. Numerical experiments are presented.

Diagnostic model for local temporal thermal change at the skin of the breast during extended application of diagnostic ultrasound.

A biophysical model is derived to account for the temporal thermal change at the skin of the breast as a result of ultrasound stimulation at the suspect lesion for seven minutes, with responses recorded using an infrared camera. Twenty-two patients were studied. The observed temporal responses for malignant cases have a different pattern from those of the benign cases studied and a mathematical model is used to investigate the controlling parameters. A new method is used to estimate the coefficients of the resulting difference equation which allows more useful diagnostic parameters to be computed than the corresponding continuous bioheat equation. The model is used to fit the experimental data. The results suggest that this method might be a rapid and noninvasive aid for distinguishing between benign and malignant breast tumours.

A new, high-resolution shock-capturing hybrid scheme of flux vector splitting-Harten's TVD

The paper presents a new high-resolution hybrid scheme combining implicit flux vector splitting with Harten's TVD, which is proved suitable for shock-capturing calculation in gasdynamics. Fluxsplitting procedures are applied to discretize the implicit part of the Euler equations whereas Harten's numerical fluxes are used to calculate the residual of steady-state solutions. It ensures good shock-capturing properties and produces sharp numerical discontinuities without oscillations. It excludes expansion shocks and leads only to physically relevant solutions. The block-line-Gauss-Seidel relaxation procedure (block-LGS) is used to solve the resulting difference equations. The time step and the CFL number are much larger than those in the linearized block-alternating-direction-implicit approximate factorization method (block-ADI). Numerical experiments suggest that the hybrid scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the steady-state solution. Hence scheme seems to lead to an effective nonoscillatory shock capturing method for steady transonic flow.

Static and dynamic snap-through of orthotropic spherical caps

This paper presents the results of a study on the axisymmetric snap-through buckling of orthotropic shallow spherical shells subjected to a total-area uniform pressure. Both static and dynamic buckling of the orthotropic caps are investigated according to the classical thin shell theory and Reissener's shallow shell assumptions. To take moderate rotation and initial imperfection into account, a set of nonlinear differential governing equations is derived in terms of the mid-surface displacement and a stress function. Two types of imperfection, and damping (in the dynamic case) are included to simulate real structures. It is seen that imperfections play an important role in the reduction of buckling loads for both static and dynamic analyses. Damping, on the other hand, increases the dynamic buckling load. The finite difference scheme is applied to represent spatial and time derivatives, and the block elimination method is used to solve the resulting difference equations.

Effectiveness of a Consistently Formulated Diffusion-Synthetic Acceleration Differencing Approach

A consistently formulated differencing approach is applied to the diffusion-synthetic acceleration of discrete ordinates calculations based on various spatial differencing schemes. The diffusion ''coupling'' equations derived for each scheme are contrasted to conventional coupling relations and are shown to permit derivation of either point- or box-centered diffusion difference equations. The resulting difference equations are shown to be mathematically equivalent, in slab geometry, to equations derived by applying Larsen's four-step procedure to the S/sub 2/ equations. Fourier stability analysis of the acceleration method applied to slab model problems is used to demonstrate that, for any S/sub n/ differencing scheme (a) the upper bound on the spectral radius of the method occurs in the fine-mesh limit and equals that of the spatially continuous case (0.22466), and (b) the spectral radius decreases with increasing mesh size to an asymptotic value <0.13135. This model problem performance is somewhat superior to that of Larsen's approach, for which the spectral radius is bounded by 0.25 in the wide-mesh limit. Numerical results of multidimensional, heterogeneous, scattering-dominated problems are also presented to demonstrate the rapid convergence of accelerated discrete ordinates calculations using various spatial differencing schemes.

Flow field calculation and its experimental verification for inertia stage of marine gas turbine air intake filter

A numerical method is presented in this paper for the solution of flow field in marine gas turbine air intake filtration channel. The flow field was successfully calculated by this method, and aerodynamic characteristics were obtained for various types of filtration channels. This work is expected to be of practical importance for the design of such filters. Upstream difference was adopted to discretize the non-conservative type N-S equation for two-dimensional, time-dependnet, viscous and incompressible flow, and the stability, convergence, accuracy and artificial viscosity of the resulting difference equation were examined. This equation can be used to calculate viscous flows with Reynolds number up to tens of thousands. Also presented in this paper is a calculation method for treating wall vortex at boundary inflection points. Careful studies show that calculation based on the difference equational and wall vortex treatment proposed here are in good agreement with experimental results.

Weierstrass elliptic difference equations

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Interference in Thin Films

The interface problem of light reflection and refraction through a thin dielectric slab is formulated as a pure BVP for the e.m. field in the slab. The resulting difference equation is solved and interference effects for quarter-wave films are discussed.

Boundedness of solutions of difference equations and application to numerical solution of Volterra integral equations of the second kind

We study the difference equations obtained when some numerical methods for Volterra integral equations of the second kind are applied to the linear test problem y( t) = 1 + ∝ 0 t ( λ + μt + vs) y( s) ds, t ⩾ 0, with fixed stepsize h. The resulting difference equations are of Poincaré type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von Neumann polynomial. This result is then used in stability analysis of reducible quadrature methods for Volterra integral equations.

A full-implicit-continuous-Eulerian (FICE) scheme for multidimensional transient magnetohydrodynamic (MHD) flows

A full-implicit-continuous-Eulerian (FICE) scheme is developed for solving multidimensional transient magnetohydrodynamic (MHD) flow problems. The resulting difference equations are solved through a single-loop iteration in which the time-advanced pressure equation is solved by using the line-by-line iteration method (Patankar, “Numerical Heat Flow,” Hemisphere, Washington, D.C., 1980). In order to keep the boundary conditions self-consistent, a new formulation of boundary conditions is developed for this two-dimensional initial boundary value magnetohydrodynamic (MHD) flow problem. The merit of this new formulation is that improved consistency and accuracy on both physical and computational boundary values are obtained when compared to earlier methods. The stipulation of the boundary conditions is based on the projected characteristic method. The boundaries in a numerical computation may be classified into the following two categories: (i) Physical boundaries, on which the number of dependent variables are to be arbitrarily specified, would be limited to the number of incoming characteristics that are projected in the n - t plane, where n is the unit normal of the boundary in question and t is time. The rest of the variables (if any) should satisfy the compatibility equations along the outgoing projected characteristics in the n - t plane. (ii) Computational boundaries, on which a related set of compatibility equations should also be satisfied. In addition, a new nonreflecting boundary condition is introduced by taking all the spatial derivative terms of dependent variables to be zero in the characteristic equations along the incoming projected characteristics in the n - t plane. A numerical example for an astrophysical fluid is given to illustrate the present algorithm and boundary conditions. In addition, the comparison between the results of using the present nonreflecting boundary condition and the two conventional ones (i.e., equivalue and linear extrapolations) is made. It shows that the nonreflecting boundary condition formulated in this paper gives much smaller (almost null) reflection after the disturbance has reached the boundary and, therefore, can provide more accurate numerical results.

Finite difference solutions of dispersive partial differential equations

The second order leapfrog method is used to discretize the linearized KdV equation which is itself a dispersive partial differential equation. The resulting difference equation is solved and analyzed in terms of its dispersion relation and propagation properties. Numerical experiments are included to illustrate some of the results obtained. Finally, a novel aliasing error which affects the propagation of wave packets is presented

Mathematical model for contact inhibited cell division

We present a mathematical model for cell growth, which takes into account cell-cell interactions and leads to non-exponential inhibited growth of number of cells. The resulting difference equation is solved and extended to a differential equation which turns out to be of a non-linear diffusion type.

Solution of the difference equation for Gauss' hypergeometric function by use of the tau method

In two previous papers the idea of the tau method to get approximate solutions of differential equations was extended to get like results for difference equations. The models treated were the difference equation for the reciprocal of the gamma function and the difference equation for the entire part of the Bessel function of the first kind. In this paper, we apply the tau method as described in the title of this paper. Some numerics are presented which indicate that the process is convergent. In this connection an ‘a posteriori’ error analysis for the Bessel function case is presented. How this analysis could be extended to the model of the present paper is discussed, but further analysis is deferred to future research.

Difference equation evaluation of the evolution of radially symmetric ultrasonic fields

A simplified difference equation technique has been developed to calculate the three‐dimensional evolution of radially symmetric complex ultrasonic fields. Localized changes in the magnitude of the field with respect to distance along the axis of propagation are first assumed to be small compared to those with respect to radial displacement. The complex Helmholtz equation in cylindrical coordinates then may be simplified by separating variables and approximating these variables by truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane from corresponding adjacent locations in the preceding plane. Stability criteria which limit both the step sizes between consecutive planes and between calculated field values in one plane are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two‐dimensional focusing and non‐focusing uniform and Gaussian profile piezoelectric transducers and theoretical and experimental data are presented. [Work supported in part by NASA and NSF.]

Navier-Stokes Computations of Turbulent Compressible Two-Dimensional Impinging Jet Flowfields

Planar mass-averaged compressible Navier-Stokes and energy equations in stream function/vorticity form are solved in conjunction with a two-equation (fc-e) turbulence model for impinging jet configurations relevant to VTOL aircraft design. The physical domain of the flow is mapped conformally into a rectangular computational region. An augmented central-difference scheme is used to preserve the diagonal dominance character of the difference equations at high Reynolds numbers. The resulting difference equations are solved by successive point relaxation. Excellent agreement with the experimental data is obtained for the airframe undersurface pressure, ground-plane pressure, and the centerline velocity decay along the jet axis. Computed turbulent kinetic energy along the jet axis, however, has a larger overshoot near the ground plane than indicated by the experimental data.