Observability of Discretized Partial Differential Equations

The situation in which one cannot solve a numerical initial value problem for lack of complete initial data is arising ever more frequently in applied sciences and engineering. The procedure of estimating the evolving solution by inserting incomplete data into a numerical model as they become available at different instants of time is called data assimilation or filtering. We show that if data generated by a linear system of partial differential equations (PDE) are inserted properly, then complete observability of the discrete numerical model is necessary and sufficient for asymptotic stability of the data assimilation process. This complete observability means that if the data had been generated by the discrete model rather than by the PDE system, then the data would define the state of the model uniquely after some finite time. Simple observability criteria for discretizations of linear, constant-coefficient PDE on periodic domains are formulated in terms of properties of the symbol of the numerical scheme. It is shown that spurious numerical dispersion can detract from observability by yielding a multivalued symbol. While observability for the advection equation demands the addition of dissipation to the leapfrog and Crank–Nicolson schemes, dissipation is not needed for second-order and sixth-order accurate box schemes.