Robert Vichnevetsky

A new stable computing method for the serial hybrid computer integration of partial differential equations

Partial differential equations involving one space dimension and time can be solved by hybrid computers using the serial (or continuous space-discrete time) method. In so doing, the continuous integration capability of the analog computer is used along the space axis while integration along the time axis is performed in a discrete fashion by making use of finite differences.

Analog/hybrid solution of partial differential equations in the nuclear industry

This article was presented at the AICA meeting in Versailles. It is printed here with the permission of Association interna tionale pour le Calcul Analogique.

Hybrid methods for partial differential equations

Historically, one may identify many of the hybrid and analog methods for partial differential equations with the &dquo;methods of lines,&dquo; which were first described in the late 1920’s and early 1930’s. Several papers published then were concerned with problems in two independent variables, in which the technique of approximation by finite differences was used to reduce the partial differential equations to systems of ordinary differential equations.20,28 The original intent of these methods was to provide a means of paper-and-pencil solution or to approxi-

Error Analysis in the Computer Simulation of Dynamic Systems: Variational Aspects of the Problem

Error analysis in the computer simulation of dynamic systems is fundamentally a variational problem. The computing errors are small variations of the computed solutions with respect to the exact solution of the differential equations being integrated. It is not surprising, therefore, that many of the mathematical tools used to perform error analysis in the computer simulation of dynamic systems are similar to those used in the variational or perturbational analysis of those systems themselves. Fundamental papers in this direction have been published previously [8], [9], [13]. The present paper, however derives the error-propagation equations in a more basic form, which makes it easier to apply such variational mathematical tools as Liapunoffs second method to analyze error stability, and Pontryagins maximum principle to study ``worst-case errors in computation.

A hybrid computer method for the analysis of time dependent river pollution problems

This paper is devoted to the description of work done in the hybrid computer simulation of polluted rivers and estuaries. Our attention in this paper is restricted to the solution of the pollutant concentration equation. The computer method used to perform the integration is essentially a continuous-space discrete-time method of lines. We have, in a previous paper, described a continuous-space-discrete-time computer method for the analysis of flows and velocities in a one-dimensional river or estuary. Hence, these two programs, which may be exercised simultaneously, must be viewed as part of the same problem, since the pollutant diffusion parameters in a river (as described in the present paper) may be derived as explicit functions of the river geometry and water flow.

Generalized finite-difference approximations f or the parallel solution of initial value problems

Methods used in analog computation for the parallel- finite-differences solution of partial differential equations have been almost universally based on the classical derivation of finite-difference approximations. That is, the space-dependence of the approximate solution is lo cally assumed to be appropriately represented by a Tay lor (or polynomial) truncated series of low1, or higher7, order. Although known to introduce higher truncation errors than the use of other functional series2,3, this re striction has been mostly accepted for reasons of con venience because no simple method for obtaining more general approximations was available in practice. We show in this paper that completely general finite- difference approximations of linear partial differential operators in space, based on functional (rather than, but including, polynomial) approximations, can be easily ob tained in a direct method involving only the inversion and multiplication of constant matrices. One of the drawbacks of the analog computer imple mentation of equations obtained by application of the method presented here is that it requires more attenu ators than the classical low-order method. However, as suggested by Deiters and Nomura3, this is alleviated in a hybrid computer by introducing at discrete time inter vals the higher-order terms as digital corrections super imposed upon an analog low-order implementation. Examples of the derivation of generalized finite-differ ence equations are shown for the representation of the Laplacian (diffusion) operator in circular (r, θ) and linear (x) coordinates.

Stability contours for the analysis of analog/digital hybrid simulation loops

The use of hybrid computers for the simulation of dynamic systems has focused interest on the ill-effects of sampling and digital execution time upon solution accuracy. Techniques have been proposed for the compensation of these effects by various authors. However, there has been a noted lack of common denominator available to compare the relative merits of these methods of compensation. As a rule, one may observe that quality criteria used to evaluate different compensation methods have been local rather than global, and the resulting compensation algorithms are therefore largely tuned to whatever quality criteria has been selected a priori.

A hybrid time mark program

The program The R (Ready) signal, indicating that the digital computer has outputted new information, is used to initialize shift register SR 1 on its leading edge and to start the operation by setting the flip flop on its trailing edge (unclocked logic is assumed throughout). SR 1 is used as a serializer; it samples, in sequence, each bit of the value outputted from the digital computer, and connects the event marker to the output of either monostable timer MT 2 tor MT 3, depending on whether the sampled bit is high or low, respectively. MT 1 generates the basic frame, at the end of which it restarts both MT 2 and MT 3, as well as itself, and advances the serializer SR 1. When all bits have been sampled (bit 7 of SR 1 goes low), the flip f p is reset and the process is stopped. If the time reference of the simulation is not generated in the digital computer, then a counter/shift-register can be used to both keep track of time and to present the bits, one at a time, for selecting between the outputs of MT 2 and MT 3.

Eastern Simulation Council meeting

program. After the welcoming address by Romeo R. Favreau, Vice President EAI Research and Computation Division, and opening remarks by Walter Brunner, Director of the Princeton Computation Center, the technical portion of the program began with a tutorial paper by Dr. Jon C. Strauss: A FORTRAN dialect of the SCi continuous system simulation language. The particular continuous system simulation language which Jon described is known as the BHSL (Basic Hytran Simulation Language), developed at EAI. He reminded those present that Simulation Councils, Inc., sponsors a National Committee whose purpose is to specify stand-

Simulation of a silicon controlled rectifier

This assumes no reverse current in the SCR. This assumption does not seriously affect the accuracy of the simulation if the SCR is carrying relatively moderate or heavy load currents. The reverse current is essentially a measure of the stored carriers in the junction. If the time derivative of the voltage across the SCR is great enough, reverse currents due to the junction capacitance can be appreciable. This is no problem to simulate and will not be discussed here. It is suggested that this capacitance be ignored for the initial simulation. If dv/dt appears excessive, the capacitance can then be added. One should not forget the trigger capacitance and possible false gating. The implementation of the boolean statement is shown in figure 2. This in itself is not sufficient. Some communication must exist between the analog and digital hardware. A simple comparator to digitize the analog signals, and relay contacts to change the analog equations are adequate communication devices. The comparator gives a logical &dquo;1&dquo;