Two Simple Numerical Methods for the Free Boundary in One-Phase Stefan Problem

Table 6

Comparison of VSG, BIM, LTUF, LTFF method, and exact solution when 𝑓(𝑡)=exp(𝑡+1).

Position 𝑥	Numerical solution				Exact solution
	VSG	BIM	LTUF	LTFF
0.00	6.389056	6.389056	6.389056	6.389056	6.389056
0.10	5.685461	5.685448	5.685467	5.685440	5.685894
0.20	5.049275	5.049261	5.049281	5.049254	5.049647
0.30	4.473626	4.473613	4.473632	4.473605	4.473947
0.40	3.952754	3.952741	3.952761	3.952734	3.953032
0.50	3.481448	3.481434	3.481454	3.481428	3.481689
0.60	3.054989	3.054975	3.054997	3.054978	3.055199
0.70	2.669112	2.669098	2.669120	2.669094	2.669296
0.80	2.319954	2.319939	2.319962	2.319937	2.320117
0.90	2.004020	2.004006	2.004030	2.004006	2.004116
1.00	1.718151	1.718137	1.718162	1.718139	1.718281
1.10	1.459484	1.459470	1.459498	1.459474	1.459603
1.20	1.225432	1.225418	1.225446	1.225425	1.225540
1.30	1.013652	1.013638	1.013667	1.013648	1.013752
1.40	0.822024	0.822010	0.822042	0.822024	0.822119
1.50	0.648631	0.648618	0.648651	0.648635	0.648721
1.60	0.491738	0.491725	0.491760	0.491746	0.491824
1.70	0.349775	0.349762	0.349799	0.349787	0.349859
1.80	0.221320	0.221308	0.221347	0.221338	0.221403
1.90	0.105089	0.105077	0.105119	0.105112	0.105171
2.00	0.000000	0.000000	0.000000	0.000000	0.000000
RMSE	0.000109	0.000107	0.000108	0.000092
L2 norm	0.001552	0.001516	0.001540	0.001315
Runtime (secs)	3.640	4.122	3.700	3.969