Transient circuit simulation for differential algebraic systems using matrix exponential

Transient simulation becomes a bottleneck for modern IC designs due to large numbers of transistors, interconnects and tight design margins. For modified nodal analysis (MNA) formulation, we could have differential algebraic equations (DAEs) which consist ordinary differential equations (ODEs) and algebraic equations. Study of solving DAEs with conventional multi-step integration methods has been a research topic in the last few decades. We adopt matrix exponential based integration method for circuit transient analysis, its stability and accuracy with DAEs remain an open problem. We identify that potential stability issues in the calculation of matrix exponential and vector product (MEVP) with rational Krylov method are originated from the singular system matrix in DAEs. We then devise a robust algorithm to implicitly regularize the system matrix while maintaining its sparsity. With the new approach, $\varphi$ functions are applied for MEVP to improve the accuracy of results. Moreover our framework no longer suffers from the limitation on step sizes thus a large leap step is adopted to skip many simulation steps in between. Features of the algorithm are validated on large-scale power delivery networks which achieve high efficiency and accuracy.