Research Article

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

Table 1

Results obtained when the test problem defined by (5.3)–(5.5) 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

11681607.373E-01β€”1.454E-01β€”
23363204.003E-011.8421.741E-028.350
36726401.254E-013.1421.224E-0314.220
4134412803.080E-024.1357.730E-0515.837
5268825607.765E-033.9674.841E-0615.970
6537651201.954E-033.9743.026E-0715.999
710752102404.892E-043.9941.891E-0816.004
821504204801.224E-043.9991.181E-0916.011
943008409603.059E-054.0007.609E-1115.519
1086016819207.648E-064.0009.848E-127.726
111720321638401.912E-064.0004.966E-121.983