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Applied Math Seminar On error bounds of
finite difference approximations to partial differential equations |
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We consider a family of spatially semi-discrete approximations, including boundary treatments, to hyperbolic and parabolic equations. We derive the dependence of the error-bounds on time as well as on mesh size. The question of the role of numerically imposed boundary conditions in the solution of parabolic and hyperbolic PDE's has been with us for many years. Many investigators have studied the effect of boundary conditions on the stability of the overall scheme, (i.e. the 'inner algorithm' + boundary conditions) In this context stability implies convergence of the scheme, at a fixed time $t$, as the mesh is refined. The question of the temporal behavior of the error was usually not considered. When constructing higher order schemes, 3rd-order accuracy and above, it turns out that it is difficult to state boundary conditions such that the overall scheme remains stable. The question then arises what happens to the overall accuracy of the numerical solution if the order of accuracy of the inner stencil, m, is is higher then the order of the boundary conditions, (m-s). This problem has been tackled by Gustafsson. His main result, for both parabolic and hyperbolic PDE's, is that if the accuracy of the extra boundary conditions, required for 'numerical closure' of the problem, are one less than that of the inner scheme, then the overall accuracy is not affected. The physical boundary conditions, however, must be approximated to the same order as the inner scheme. In the present work we had considered a form of differentiation matrices, both hyperbolic and parabolic, which represent a fairly wide family of boundary condition formulation plus central inner schemes. We investigate the dependence of the error on time as well as on mesh size. The main results are as follows: 1. In the hyperbolic case, the overall convergence rate is of order min(m,m-s+1) for all s, in agreement with the results given by Gustafsson. For s=0,1, the temporal bound of the error behaves as sqrt{t} for t<1, and bends over smoothly to a linear bound as t increases. For s>=2, the temporal behavior is of the order of sqrt(t) for all t>=0. 2. In the parabolic case, the overall convergence rate is of order m if s=0,1, and m-s+3/2 if s>=2. The error is uniformly bounded independent of t for all t>=0. Numerical experiments, demonstrate the validity of these bounds. In fact, in the parabolic case, the numerical convergence rate is (m-s+2), s>=2, exceeding the prediction which is only an upper bound on the error. This is a joint work with S. Abarbanel and B. Gustafsson. |