I. Introduction
Analog computing can accelerate certain special classes of computational problems involving partial differential equations (PDEs) defined from continuous-time systems [1]–[11]. Most physics based simulations involve either linear or nonlinear PDE systems. Electromagnetics, which is an extremely important branch of physics, is completely described by Maxwell's equations, which are in turn, first order linear PDEs [12]–[14]. Therefore, there has been immense efforts in the computational electromagnetics community to simulate electromagnetic models by first descretizing the Maxwell's equations using a staggered computational grid and then solving by digital computers using software [15]–[19]. However, Maxwell's equations are themselves time-continuous, and therefore allow a natural fit for linear analog computing systems using simulation models based on spatially-discrete-time-continuous update equations. We demonstrate the concept by proposing a simplified analog computer that solves the spatio-temporal wave-equation using an analog array processor that operates in continuous-time mode. We have developed two methods producing mathematical models, one of which has been realized using analog circuits. A low-frequency implementation based on operational amplifiers (op-amps) is used to demonstrate the new analog algorithms, with RF extension part of on-going work not covered in this paper.
Systolic array architecture of the second order continuous-time PDE solver. Block diagram of the internal module (IM) derived from (b) method 1 and (c) method 2.