The finite precision computation and the nonconvergence of difference scheme
This paper identifies a fundamental limitation of finite precision computation in numerical analysis, challenging the practical validity of the Lax equivalence theorem for real-world computing.
The authors show that round-off error can break the consistency of difference schemes, leading to nonconvergence of computed results to analytical solutions even for stable schemes satisfying the Lax equivalence theorem. An experiment with a conservation scheme solving a linear differential equation demonstrates nonconvergence as time step decreases.
The authors show that the round-off error can break the consistency which is the premise of using the difference equation to replace the original differential equations. We therefore proposed a theoretical approach to investigate this effect, and found that the difference scheme can not guarantee the convergence of the actual compute result to the analytical one. A conservation scheme experiment is applied to solve a simple linear differential equation satisfing the LAX equivalence theorem in a finite precision computer. The result of this experiment is not convergent when time step-size decreases trend to zero, which proves that even the stable scheme can't guarantee the numerical convergence in finite precision computer. Further the relative convergence concept is introduced.