This dissertation presents two circuit simulation techniques to accelerate the simulation time for time-domain transient circuit simulation and circuit thermal analysis. Matrix exponential method is one of the state-of-the-art methods for millionth-order time-domain circuit simulations due to its explicit nature and global stability. The matrix exponential is commonly computed by Krylov subspace methods, which become inefficient when the circuit is stiff, namely when the time constants of the circuit differ by several orders. The truncated Faber series is suitable for accurate evaluation of the matrix exponential even under a highly stiff system matrix arising from practical circuits. Experiments have shown that the proposed approach is globally stable, highly accurate and parallelizable, and avoids excessive memory storage demanded by Krylov subspace methods. Another major issue in circuit simulation is thermal circuit analysis. The use of Hierarchical matrix (H-matrix) in the efficient finite-element-based (FE-based) direct solver implementation for both steady and transient thermal analyses of three-dimensional integrated circuits (3D ICs) is proposed. H-matrix was shown to provide a data-sparse way to approximate the matrices and their inverses with almost linear space and time complexities. This is also true for FE-based transient analysis of thermal parabolic partial differential equations (PDEs). Specifically, the stiffness matrix from a FE-based steady and transient thermal analysis can be represented by H-matrix without approximation, and its inverse and Cholesky factors can be evaluated by H-matrix with controlled accuracy. This thesis shows that the memory and time complexities of the solver are bounded by O(k_1NlogN) and O(K_1^2Nlog〖log〗^2N), respectively, for very large scale thermal systems, where k1 is a small quantity determined by accuracy requirements and N is the number of unknowns in the system. Numerical results validate and demonstrate the effectiveness of the proposed method in terms of predicted theoretical scalability. / published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy