Numerical methods as applied to linear and nonlinear wave equations with solutions having different levels of smoothness

A variety of numerical methods are applied to solving the wave equations u_tt = u_xx and u_tt = u_xx - u^3 on the bounded interval x=[-1,1], with Dirichlet boundary conditions u=0 at the endpoints, and initial conditions u = f(x) and u_t = 0. For the linear equation the d'Alembert solution on an infinite interval can be adapted to yield an analytical solution also on the bounded interval. This d'Alembert solution shows that discontinuities can develop in the second (or higher even) derivatives, even if the initial condition f(x) itself is infinitely differentiable. The exact solution of our nonlinear wave equation is unknown, but the discontinuities presented in the linear equation are the same for this nonlinear equation as they share the same characteristics. On the other hand, this nonlinear equation also has an analytic result, namely energy conservation. In short, the exact solution of the linear/nonlinear problems have these different properties:
1: The solution is infinitely smooth at some special times, t=0,2,4,6,8,etc., only.
2: The energy is conserved, thus, the overall solution amplitude remains a constant.
A major focus in this thesis will be on how well following various numerical methods cope with three different initial conditions carefully chosen to have different levels of smoothness in their later evolution.
The numerical methods used are the spectral methods (SM) and the finite element methods (FEM).
Many time stepping methods are studied: Euler's, modified Euler, implicit Euler and trapezoidal methods, as well as the exponential methods ETD1 and ETD2RK.