Exponential integrators for the investigation of stability of nonlinear waves

Abstract

Nonlinear wave equations have long been acknowledged for their key role in describing wave phenomena under different circumstances; in particular, they have widespread application in oceanography. This thesis considers the Korteweg-de Vries (KdV) and its extensions, such as the Gardner equation, the forced Korteweg-de Vries (fKdV) equation and the forced Gardner equation as an identifiable example of nonlinear wave equations that balances quadratic, cubic nonlinearity and dispersion. We propose numerical methods to investigate these nonlinear wave equations in the weak dispersion limit, utilising the pseudospectral method for spatial discretisation and fourth-order semi-implicit time integration. We develop an efficient variable step-size approach using the conservation of energy (Hamiltonian) to monitor the accuracy of these methods and determine the optimal step-size. We investigate the robustness of these proposed numerical methods for a range of nonlinear wave equations and various ranges of initial conditions, and that the wave resistance is essential to understanding the nonlinearity of the fKdV equation and forced Gardner equation. We compare the error and Hamiltonian against the cost for each numerical method with fixed and variable step-size approaches. Numerical results show that the Hamiltonian accuracy behaves equivalently to the accuracy obtained for the exact solution. We also evaluate the efficiency of the numerical methods by comparing fixed step-size and variable step-size approaches. The fourth-order linearly implicit Runge– Kutta method is found to be the most efficient of the considered methods, whereas the Fourier split-step method is the least efficient method. For two exponential time differencing methods the fixed step-size approach is more efficient than the variable step-size approach. The variable step-size approach with the fourth-order linearly implicit Runge– Kutta method is generally comparable to the fixed step-size approach, although the variable step-size approach allows the user to specify the desired accuracy. Furthermore, we examine the effect of one parameter for the algorithm, the safety factor, on the variable step-size approach for the fKdV equation. We numerically analyse the unsteady solutions to the negative forced KdV equation for the zero initial condition and weak dispersion. We use the wave resistance to classify the types of solution regimes and the wave resistance power spectrum to identify the dominant frequencies of the solutions. We identify three main regimes: supercritical, transcritical and subcritical. Each regime contains classical, intermediate and weak dispersion subregimes. Moreover, we discuss the nature of instabilities over the topography for weak dispersion. The solution regimes for the positive and negative forced Gardner equation, with zero initial condition and in the weak dispersion limit, are also determined. We compare the theoretical and numerical results for the positive forced Gardner equation, and examine a twin supercritical leaps regime, which was not previously observed. We obtain numerical results and examine the bifurcation point for the negative forced Gardner equation, where the cubic nonlinearity becomes significant. In contrast, there is no evidence of twin supercritical leap regimes for the negative forced Gardner equation. However, we observe a kink soliton behaviour in the approach-controlled and reversed constriction-controlled regimes for a negative forced Gardner equation.