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

Table 3

Results obtained when the test problem defined in Section 5.4 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 “Error”. In row 𝑠,𝑠=2,3,…,11, the ratios of the errors in this row and in the previous row are given in the columns under “Ratio”.

			Direct solution		Richardson Extrapolation
No.	𝑁𝑡	𝑁𝑥	Error	Ratio	Error	Ratio
1	168	160	1.353E-01	—	4.978E-02	—
2	336	320	7.687E-02	1.760	2.761E-02	1.803
3	672	640	4.424E-02	1.737	1.551E-02	1.780
4	1344	1280	2.555E-02	1.732	8.570E-03	1.810
5	2688	2560	1.636E-02	1.561	4.590E-03	1.867
6	5376	5120	1.051E-02	1.552	2.318E-03	1.980
7	10752	10240	5.551E-03	1.899	1.188E-03	1.951
8	21504	20480	2.921E-03	1.900	6.575E-04	1.807
9	43008	40960	2.644E-03	1.105	2.379E-04	2.746
10	86016	81920	1.619E-03	1.633	1.501E-04	1.585
11	172032	163840	1.145E-03	1.414	2.787E-05	4.941