This thesis develops a study for the frequency warping introduced by time-domain methods. The work in this study focuses first on the time-domain methods used in the classical SPICE engine, that is the Backward Euler, the Trapezoidal Rule and the Gear's methods. Next, the thesis considers the newly developed high-order method based on the Obreshkov formula. This latter method was proved to have the A-stability and L-stability properties, and is therefore robust in circuit simulation. However, to the best of the author's knowledge, a mathematical study for the frequency warping introduced by this method has not been developed yet. The thesis therefore develops the mathematical derivation for the frequency warping of the Obreshkov-based method. The derivations produced reveal that those methods introduce much smaller warping errors than the traditional methods used by SPICE. In order to take advantage of the small warping error, the thesis develops a shooting method framework based on the Obreshkov-based method to compute the steady-state response of nonlinear circuits excited by periodical sources. The new method demonstrates that the steady-state response can be constructed with much smaller number of time points than what is typically required by the classical methods.

Flooding has always been and is still today a disastrous event with agricultural, infrastructural, economical and not least humanitarian ramifications. Understanding the behaviour of floods is crucial to be able to prevent or mitigate future catastrophes, a task which can be accomplished by modelling the water flow. In this thesis the finite element method is employed to solve the shallow water equations, which govern water flow in shallow environments such as rivers, lakes and dams, a methodology that has been widely used for flooding simulations. Alternative approaches to model floods are however also briefly discussed. Since the finite element method suffers from numerical instabilities when solving nonlinear conservation laws, the shallow water equations are stabilised by introducing a high-order nonlinear artificial viscosity, constructed using a multi-mesh strategy. The accuracy, robustness and well-balancedness of the solution are examined through a variety of benchmark tests. Finally, the equations are extended to include a friction term, after which the effectiveness of the method in a real-life scenario is verified by a prolonged simulation of the Malpasset dam break.

In this thesis, we first propose a new scalar auxiliary variable (SAV) approach for general dissipative nonlinear systems. This new approach is half computational cost of the original SAV approach, can be extended to high order unconditionally energy stable backward differentiation formula (BDF) schemes and not restricted to the gradient flow structure. Rigorous error estimates for this new SAV approach are conducted for the Allen-Cahn and Cahn-Hilliard type equations from the BDF1 to the BDF5 schemes in a unified form. As an application of this new approach, we construct high order unconditionally stable, fully discrete schemes for the incompressible Navier-Stokes equation with periodic boundary condition. The corresponding error estimates for the fully discrete schemes are also reported. Secondly, by combining the new SAV approach with functional transformation, we propose a new method to construct high-order, linear, positivity/bound preserving and unconditionally energy stable schemes for general dissipative systems whose solutions are positivity/bound preserving. We apply this new method to second order equations: the Allen-Cahn equation with logarithm potential, the Poisson-Nernst-Planck equation and the Keller-Segel equations and fourth order equations: the thin film equation and the Cahn-Hilliard equation with logarithm potential. Ample numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.