The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations
It is shown that problem sensitivity and finite precision arithmetic can combine to produce false numerical solutions to steady state problems. Although one might guess that it is possible to construct pathological examples with this property, it is somewhat remarkable that this phenomenon can occur for a ``simple'' Burgers' equation. In addition, in the space of antisymmetric \(L^2 \) functions \(L_{AS}^2 (0,1)\), the authors have shown that the steady state Burgers' equation has a unique solution \((v(x)= 0)\), and yet, discretized versions of this equation can yield nonunique solutions. More importantly, for sufficiently large initial data, marching schemes will converge to this discrete (yet false) solution. The authors also presented a sensitivity and stability analysis that provides insight into these numerical difficulties. One implication of these results is that more analysis needs to be done on recent ``numerical based proofs'' of nonuniqueness. In particular, it is established that in a perfectly reasonable model, numerical computations that yield nonunique discrete stationary solutions do not imply anything about nonuniqueness for the continuous boundary value problem. It is important to again emphasize that it is finite precision arithmetic that causes these false solutions. Therefore, even converging an algorithm to machine roundoff error will not eliminate the difficulty. Finally, it is observed that the false numerical solutions can differ by orders of magnitude from real stationary solutions. Therefore, in such cases, cascading the numerical solutions into an optimization or control algorithm can produce bad designs.
scientific article; zbMATH DE number 2093859
scientific article