Showing papers on "Finite difference method published in 1981"
TL;DR: The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one as mentioned in this paper, which readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum.
2,042 citations
TL;DR: In this paper, the relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated, and applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behaviour of parasitic waves, and the stability analysis of initial boundary value problems.
Abstract: The relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated. Applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behavior of parasitic waves, and the stability analysis of initial boundary-value problems.
477 citations
TL;DR: In this article, the general problem of finite differencing the diffusion equation on a two-dimensional Lagrangian hydrodynamic mesh is discussed and a set of general criteria is developed.
131 citations
TL;DR: The finite-difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
Abstract: A current, linear, two-dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the model equations. The finite-difference method is used to discretize Laplace's equation over a rectangular region with specified boundary conditions. The resulting matrix equation for fluid pressure is solved by using a Gaussian block-elimination technique. Numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes. The finite difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
118 citations
TL;DR: In this article, a new method to solve the Boltzmann equation is proposed, based on concepts from the kinetic theory of gases. But it is not shown that the method can offer significant advantages over standard finite difference methods for certain problems, such as the Riemann shock-tube problem.
115 citations
TL;DR: In this article, it was shown that substantial azimuthal current can be driven provided that the amplitude of the rotating magnetic field is greater than a certain threshold value which depends on the plasma resistivity.
Abstract: The generation of a steady azimuthal current in a cylindrical plasma column using a rotating magnetic field is numerically investigated. The mixed initial-boundary-value problem is solved using a finite difference method. It is shown that substantial azimuthal current can be driven provided that the amplitude of the rotating magnetic field is greater than a certain threshold value which depends on the plasma resistivity.
83 citations
TL;DR: In this article, it was shown that failure to take these singularities into account leads to large errors in the finite-difference solution of the time-domain electromagnetic field equations.
Abstract: When the electromagnetic-field equations are solved in a region with a corner, singularities in the field or in its spatial derivatives will be present at these corners. These singularities cause the load truncation error in a finite-difference approximation of the field equations to be unbounded. In this paper it is shown that failing to take these singularities into account leads to large errors in the finite-difference solution of the time-domain electromagnetic-field equations. A simple method is described to account for these singularities while retaining the simplicity of the finite-difference formulation. Numerical results are given that demonstrate the accuracy obtained when our technique is used.
80 citations
TL;DR: In this article, the pseudospectral-Chebyshev method is shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems.
Abstract: The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.
68 citations
TL;DR: In this article, a numerical procedure has been developed to investigate the interaction of thermal radiation with conduction and convection in thermally developing gas-particulate suspension flow through a circular tube.
64 citations
64 citations
TL;DR: Two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified.
Abstract: In this paper we construct and analyze high order finite difference discretizations of a class of elliptic partial differential equations. In particular, two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified. The use of compact nine-point formulas of positive type admits both fast direct methods and standard iterative methods for the solution of the resulting systems of linear equations. Extensions yielding sixth order accuracy for the Helmholtz equation and fourth order accuracy for a more general operator are given. Finally, numerical results demonstrating the effectiveness of the discretizations for a wide range of problems are presented.
TL;DR: A method is described which, through the introduction of a different set of network variables, significantly reduces the size of the original system, avoids the need to compute pressures, and produces velocities that are exactly discrete divergence free.
TL;DR: In this article, a technique for moving the mesh points in physical space in order to reduce the error in the computed asymptotic solution relative to that obtained using a fixed mesh is presented.
Abstract: Coordinate system selection is an important consideration in the asymptotic numerical solution of any fluid-flow or heat transfer problem. This paper uses a new technique that provides a simple way of moving the mesh points in physical space in order to reduce the error in the computed asymptotic solution relative to that obtained using a fixed mesh. Applications to fluid-flow problems are presented, including boundary layer flow and inviscid supersonic flow over cylinders, and wedges with associated detached shocks. The treatment of curved boundaries, stationary and nonstationary boundaries, and systems of PDE's is discussed. Significant error reductions are demonstrated.
TL;DR: In this article, a phenomenological transmission line corona model is proposed to accommodate the diverse aspects of steady-state and transient power and energy losses and change in capacitance, and the model conforms with available experimental data in the literature.
Abstract: A phenomenological transmission line corona model is proposed in this paper which accommodates the diverse aspects of steady-state and transient power and energy losses and change in capacitance. Single phase line sections have been represented by a nonlinear model circuit which changes its state depending on the present and the previous values of the voltage. Propagation with corona is simulated by cascading identical model circuits and the nonlinear transmission line equations are solved by a finite difference method. The charge-potential characteristics of line sections and the nonlinear propagation characteristics of voltage surges computed numerically from our model conform with available experimental data in the literature.
TL;DR: In this paper, a methodology for the numerical prediction of 2D and 3D unsteady, in viscid, compressible flows is presented, where primitive Euler equations are recast in terms of compatibility equations on characteristic surfaces.
Abstract: A methodology for the numerical prediction of 2-D and 3-D unsteady, in viscid, compressible flows is presented. The primitive Euler equations are recast in terms of compatibility equations on characteristic surfaces. In such a way the evolution in time of the flow properties is described explicitly as the interaction of signals corresponding to the physical wave-propagation phenomenon. The equations are discretized through a finite-difference method where the proper domain of dependence of each computed point is preserved, by approximating the space-derivatives with one-sided differences, according to the velocity of propagation of signals along bicharacteristics. Results of the application of the proposed method to subsonic and transonic flow past airfoils are shown and compared with different methods.
TL;DR: In this paper, approximate solutions of effectiveness factors are systematically compared with numerical solutions for a sphere model with symmetrical boundary conditions and n th order kinetics, and the finite difference method with successive-over-relaxation is adopted for the computations.
TL;DR: In this article, a simple and efficient finite-difference technique using the generalized finitedifference (GFD) discretization is presented for two-dimensional heat transfer problems of irregular geometry, where a finite number of nodal points are distributed in the problem domain.
Abstract: A simple and efficient finite-difference technique using the generalized finite-difference (GFD) discretization is presented for two-dimensional heat transfer problems of irregular geometry. A finite number of nodal points are distributed in the problem domain. At every interior node the spatial derivatives of a field equation are approximated by functional values at neighboring nodes after introducing a family of shape functions for the dependent variables. The resulting simultaneous algebraic equations are solved in a usual manner. The results of two examples, a steady-state heat conduction and a steady natural convection problem, are compared with results of the finite-element and conventional finite-difference method, respectively. The present study demonstrates that, if well implemented, this method will become a handy yet efficient tool for solutions to any field problems since its mathematical concept is simple and the problem formulation is straightforward.
TL;DR: In this article, it was shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain.
Abstract: It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.
TL;DR: In this paper, sufficient conditons are given for the chaotic behaviour of difference equations defined in terms of continuous mappings in Rn. These conditions are applicable to both difference equations with snap-back repellors and with saddle points.
Abstract: Sufficient conditons are given for the chaotic behaviour of difference equations defined in terms of continuous mappings in Rn. These conditions are applicable to both difference equations with snap-back repellors and with saddle points. They are applied here to the twisted-horseshoe difference equation of Guckenheimer, Oster and Ipaktchi.
Dissertation•
01 Aug 1981
TL;DR: The numerical behavior of a modified orthogonal collocation method, as applied to the transport equations, can be examined through the use of a Fourier series analysis.
TL;DR: In this article, a unified method for numerical solution of nonlinear ordinary and partial differential equations (elliptic and parabolic) was developed, where the discretization error was estimated from the difference of difference quotients of a family of difference formulae.
01 Jan 1981
TL;DR: In this article, the nonlinear modified equation approach is taken to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one-and two-dimensional equations of gas dynamics.
Abstract: The nonlinear modified equation approach is taken in this paper to analyze the generalized Lax-Wendroff explicit scheme approximation to the unsteady one- and two-dimensional equations of gas dynamics. Three important applications of the method are demonstrated. The nonlinear modified equation analysis is used to (1) generate higher order accurate schemes, (2) obtain more accurate estimates of the discretization error for nonlinear systems of partial differential equations, and (3) generate an adaptive mesh procedure for the unsteady gas dynamic equations. Results are obtained for all three areas. For the adaptive mesh procedure, mesh point requirements for equal resolution of discontinuities were reduced by a factor of five for a 1-D shock tube problem solved by the explicit MacCormack scheme.
TL;DR: A numerical simulation procedure for predicting the response of a single span transmission line to strong turbulent winds is described in this article, where wind velocities are generated using a Monte Carlo technique based on an inverse fast Fourier transform; the equations of motion of the line are then solved numerically using a finite difference method.
TL;DR: In this paper, the Implicit MHD Program (IMP) was developed to solve these equations numerically by the method of finite differences on an Eulerian mesh, making the code applicable to a variety of coordinate systems.
TL;DR: In this article, a method for computing the electrical DC potential of an arbitrary three-dimensional resistivity structure using a finite difference method was developed for interpretation of geoelectrical soundings with controlled point sources over a laterally inhomogeneous medium.
Abstract: A method has been developed for computation of the electrical DC potential of an arbitrary three-dimensional resistivity structure using a finite difference method. The threedimensionality is necessary for interpretation of geoelectrical soundings with controlled point sources over a laterally inhomogeneous medium. Lateral inhomogeneities should be considered in resistivity soundings with large layouts. The results obtained with the described method permit a more realistic representation of geological features.
The resolution of the method is determined by the number of elements in the resistivity network. The problem of core memory space has been resolved by using random access disc files. The results computed using a Fortran program are in good agreement with analytically obtained solutions.
TL;DR: In this paper, the transonic cascade flow is calculated with an efficient and flexible Galerkin Finite Element method applied to the full potential equation in Artificial Compressibility form, and the important problem of obtaining a well-constructed mesh is solved satisfactorily by automatic grid generation based on the solution of two elliptic partial differential equations.
Abstract: The transonic cascade flow is calculated with an efficient and flexible Galerkin Finite Element method applied to the full potential equation in Artificial Compressibility form. Some of the typical advantages of finite element techniques are demonstrated such as the use of higher order discretization with biquadratic elements besides the classical bilinear second order accurate element, automatic treatment of the body fitted mesh due to the locally defined isoparametric mapping, easy and exact introduction of arbitrary Neumann boundary conditions along curvilinear boundaries. On the other hand, the conceptual simplicity and efficiency of the finite difference methods based on the same equation and developed for external flows are fully maintained by the use of line relaxation or approximate factorization for the iterative solution algorithm, eventually combined with a multigrid approach. The important problem of obtaining a well-constructed mesh is solved satisfactorily by automatic grid generation based on the solution of two elliptic partial differential equations. Calculations are presented and compared with experimental data for both compressor and turbine cascade flows containing shocks.
TL;DR: In this paper, an implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws, which is a two-level scheme which is unconditionally stable and nondissipative.
Abstract: An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.
TL;DR: In this article, an effective numerical approach to the solution of the two-dimensional inverse geomagnetic induction problem using the linearization method is presented, based on Marquardt's algorithm, for which the solution and the partial derivatives of this solution with respect to the electrical parameters of the medium are computed by the finite difference method.
Abstract: An effective numerical approach to the solution of the two-dimensional inverse geomagnetic induction problem using the linearization method is presented. The numerical realization of the inversion is based on Marquardt's algorithm, for which the solution of the direct problem and the partial derivatives of this solution with respect to the electrical parameters of the medium are computed by the finite difference method. Theoretical models are studied and numerical results are presented.