The optimal accuracy of difference schemes

Source abstract

We consider difference approximations to the model hyperbolic equation u t = u x {u_{t}} = {u_x} which compute each new value U ( x , t + Δ t ) U(x,t + \Delta t) as a combination of the known values U ( x − r Δ x , t ) , … , U ( x + s Δ x , Δ t ) U(x - r\Delta x,t),\ldots ,U(x + s\Delta x,\Delta t) . For such schemes we find the optimal order of accuracy: stability is possible for small Δ t / Δ x \Delta t/\Delta x if and only if p ⩽ min { r + s , 2 r + 2 , 2 s } p \leqslant \min \{ {r + s,2r + 2,2s} \} . A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations P ( z ) / Q ( z ) P(z)/Q(z) to z λ {z^\lambda } near z = 1 z = 1 , and we find an expression for the difference | Q | 2 − | P | 2 |Q{|^2} - |P{|^2} ; this allows us to test the von Neumann condition | P / Q | ⩽ 1 |P/Q| \leqslant 1 . We also determine the number of zeros of Q Q in the unit circle, which decides whether the implicit part is uniformly invertible.