Numerical simulation of Poincaré waves using central and non-central compact schemes

Several numerical methods are employed to solve the linearized shallow-water equations describing the propagation of Poincaré waves within a one-dimensional finite domain. An analytical solution to the problem, set off by a discontinuous step like elevation, is known and allows to assess the accuracy and robustness of each method and in particular, their ability to capture the traveling discontinuities without generating spurious oscillations.
    The present work examines and applies the central and non-central compact finite difference schemes for the numerical solution of the governing equations of Poincaré waves. Undoubtedly, the central and non-central compact spatial discretization methods have higher numerical accuracy than the central second-order method, and in places where there is an exact solution, the compact methods have shown that these methods are stable under various applied boundary conditions and three-diagonal and five-diagonal forms can be used according to possible limitations. The fourth-order central compact, the third-order and the fifth-order non central compact methods are employed to carry out the spatial differencing of the governing equations and a fourth-order Runge-Kutta method is used for the temporal discretization. The Runge-Kutta time discretization method of the fourth order is a four-step method. In each step, a value for an assumed function is calculated in an intermediate time step, and in the next step, in the same time step, this value is modified.
    In this research, first, the one-dimensional advection equation, which has an analytical solution, is discretized using the above methods, and the performance and numerical accuracy of the methods are measured. Then, the governing equations of Poincaré waves are numerically solved using the mentioned methods and the results are compared for two initial conditions with discontinuous points. The initial condition of the step function is a smooth condition that produces spurious oscillations but the initial condition of the hyperbolic tangent is a sloping condition in the corners, which produces less oscillations. Finally, the numerical solutions of the central and non-central compact methods are compared with each other and the results are analyzed.
    The central and non-central compact methods work well in detecting and identifying the traveling discontinuities. Among the used methods, the non-central compact method of the fifth order has better performance. Moreover, it has a lower error and a higher numerical accuracy. However, with the increase of grid points, the computational cost of this method increases drastically because the fifth-order non-central compact method is a five-point method, and the matrix of their coefficients forms a five-diagonal matrix which has a great impact on the computational time.