Research Article

Solving Advection Equations by Applying the Crank-Nicolson Scheme Combined with the Richardson Extrapolation

Table 2

Results obtained when the test problem defined by (5.7)–(5.9) is handled directly by the Crank-Nicolson scheme and by using the combination of the Crank-Nicolson scheme and the Richardson Extrapolation. The numerical errors calculated by (5.1) and (5.2) are given in the columns under β€œError”. In row 𝑠 , 𝑠 = 2 , 3 , … , 1 1 , the ratios of the errors in this row and in the previous row are given in the columns under β€œRatio”.

Direct solutionRichardson Extrapolation
No.   𝑁 𝑑     𝑁 π‘₯  ErrorRatioErrorRatio

11681607.851E-01β€”1.560E-02β€”
23363202.160E-013.6351.227E-0312.713
36726405.317E-024.0621.072E-0411.432
4134412801.327E-024.0071.150E-059.333
5268825603.319E-033.9971.193E-069.641
6537651208.299E-044.0001.478E-078.071
710752102402.075E-044.0001.618E-089.136
821504204805.187E-054.0001.965E-098.233
943008409601.297E-054.0002.387E-108.233
1086016819203.242E-064.0003.241E-117.365
111720321638408.104E-074.0001.267E-112.557