Fourier analysis of the Eulerian–Lagrangian least squares collocation method

A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian–Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C¹-continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one-dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability. Unconditional stability for time-stepping weights greater than or equal to 0-5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δx. As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher-wave-number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank–Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.