An adaptive wavelet method for nonlinear circuit simulation

The advance of very large scale integrated (VLSI) systems has been continuously challenging today´s circuit simulators in both computational speed and stability. A novel numerical method, the fast wavelet collocation method (FWCM), was first proposed to explore a new direction of circuit simulation. The FWCM uses a totally different numerical means from the classical time-marching or frequency-domain methods and has demonstrated several superior computational properties, such as uniform error distribution and better computational stability, as compared to that provided by the conventional simulation methods. The foundation for using wavelets to expand the solution of ordinary differential equations (ODEs) has been laid previously where linear systems were computed. However, it has not been studied in detail how to effectively apply the FWCM to solving nonlinear systems. In this paper, we explore the iterative and adaptive schemes which extend the FWCM to nonlinear systems. The proposed adaptive procedures mainly address the method of linearization of nonlinear terms after the unknown vector function is expanded into wavelet basis functions. We implemented two different adaptive schemes, multilevel adaptive and multiinterval adaptive, and evaluated their advantages and disadvantages. It is shown that the FWCM can handle nonlinear systems very efficiently with an accuracy as high as O(h⁴) for the solution and fast mapping between the values of the function and their wavelet expansion coefficients in, at most, O(N log N) operations, where h is the discrete time-interval length and N is the total number of collocation points. Furthermore, the derivatives of the unknown function can be calculated with an accuracy of O(h³) in O(N log N) operations. Numerical results are presented which match well with the SPICE simulation