Numerical solutions of Parabolic problem using and ... - Staff.city.ac.uk
Numerical solutions of Parabolic problem using and ... - Staff.city.ac.uk
Numerical solutions of Parabolic problem using and ... - Staff.city.ac.uk
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1<br />
Solving a <strong>Parabolic</strong> PDE <strong>Numerical</strong>ly<br />
Here we will find an approximation to φ(x, t), the solution <strong>of</strong><br />
∂φ<br />
∂t = ∂2 φ<br />
∂x 2,<br />
with φ(x, 0) = 0, 0 < x < 1 <strong>and</strong> φ(0, t) = φ(1, t) = 1.0, t ≥ 0.<br />
We will show the results in general for x = 0.0, 0.2, . . . , 1.0 <strong>and</strong> for<br />
t = 0.0, 0.05, . . . , 0.50, even though the steps in the x <strong>and</strong> t directions may be a<br />
good deal smaller.<br />
The ex<strong>ac</strong>t solution to the partial differential equation is<br />
u(x, t) = 1 −<br />
∞∑<br />
n=1<br />
4<br />
(2n − 1)π sin(2n − 1)πx e−(2n−1)2 π 2t .
2<br />
Here we will look at two finite difference schemes:<br />
• An explict scheme<br />
• The implicit Crank-Nicolson scheme.
3<br />
Solving the <strong>Parabolic</strong> PDE Using an Explicit Method<br />
We will use the scheme<br />
φ i,j+1 = rφ i−1,j + (1 − 2r)φ i,j + rφ i+1,j<br />
where r = k/h 2 , with x = hi <strong>and</strong> t = jk.
4<br />
Solution with r = 1/4. First 4 steps.<br />
h = 0.200000003, k = 0.0100000007<br />
Using 5 sp<strong>ac</strong>e steps. Need 50 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
1 t=0.010 1.0000 0.2500 0.0000 0.0000 0.2500 1.0000<br />
2 t=0.020 1.0000 0.3750 0.0625 0.0625 0.3750 1.0000<br />
3 t=0.030 1.0000 0.4531 0.1406 0.1406 0.4531 1.0000<br />
4 t=0.040 1.0000 0.5117 0.2188 0.2188 0.5117 1.0000
5<br />
Solution with r = 1. First 4 steps.<br />
h = 0.200000003, k = 0.0400000028<br />
Using 5 sp<strong>ac</strong>e steps. Need 12 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
1 t=0.040 1.0000 1.0000 0.0000 0.0000 1.0000 1.0000<br />
2 t=0.080 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000<br />
3 t=0.120 1.0000 2.0000 0.0000 0.0000 2.0000 1.0000<br />
4 t=0.160 1.0000 ****** 2.0000 2.0000 ****** 1.0000<br />
Note: The ****** indicates a number outside the range 0.0000 to 9.9999. In this<br />
case both are −1.
6<br />
Solution with r = 1/4, up to t = 1/2<br />
h = 0.200000003, k = 0.0100000007<br />
Using 5 sp<strong>ac</strong>e steps. Need 50 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
5 t=0.050 1.0000 0.5605 0.2920 0.2920 0.5605 1.0000<br />
10 t=0.100 1.0000 0.7348 0.5708 0.5708 0.7348 1.0000<br />
15 t=0.150 1.0000 0.8394 0.7402 0.7402 0.8394 1.0000<br />
20 t=0.200 1.0000 0.9028 0.8427 0.8427 0.9028 1.0000<br />
25 t=0.250 1.0000 0.9411 0.9048 0.9048 0.9411 1.0000<br />
30 t=0.300 1.0000 0.9644 0.9423 0.9423 0.9644 1.0000<br />
35 t=0.350 1.0000 0.9784 0.9651 0.9651 0.9784 1.0000<br />
40 t=0.400 1.0000 0.9869 0.9789 0.9789 0.9869 1.0000<br />
45 t=0.450 1.0000 0.9921 0.9872 0.9872 0.9921 1.0000<br />
50 t=0.500 1.0000 0.9952 0.9923 0.9923 0.9952 1.0000
7<br />
Ex<strong>ac</strong>t solution up to t = 1/2<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
t=0.050 1.0000 0.5384 0.2637 0.2637 0.5384 1.0000<br />
t=0.100 1.0000 0.7210 0.5487 0.5487 0.7210 1.0000<br />
t=0.150 1.0000 0.8297 0.7245 0.7245 0.8297 1.0000<br />
t=0.200 1.0000 0.8960 0.8318 0.8318 0.8960 1.0000<br />
t=0.250 1.0000 0.9365 0.8973 0.8973 0.9365 1.0000<br />
t=0.300 1.0000 0.9613 0.9373 0.9373 0.9613 1.0000<br />
t=0.350 1.0000 0.9763 0.9617 0.9617 0.9763 1.0000<br />
t=0.400 1.0000 0.9856 0.9766 0.9766 0.9856 1.0000<br />
t=0.450 1.0000 0.9912 0.9857 0.9857 0.9912 1.0000<br />
t=0.500 1.0000 0.9946 0.9913 0.9913 0.9946 1.0000
8<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.200000003, k = 0.0100000007<br />
Using 5 sp<strong>ac</strong>e steps. Need 50 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
5 t=0.050 0.0000 2.2193 2.8319 2.8319 2.2193 0.0000<br />
10 t=0.100 0.0000 1.3747 2.2132 2.2132 1.3747 0.0000<br />
15 t=0.150 0.0000 0.9707 1.5705 1.5705 0.9707 0.0000<br />
20 t=0.200 0.0000 0.6740 1.0905 1.0905 0.6740 0.0000<br />
25 t=0.250 0.0000 0.4607 0.7455 0.7455 0.4607 0.0000<br />
30 t=0.300 0.0000 0.3111 0.5034 0.5034 0.3111 0.0000<br />
35 t=0.350 0.0000 0.2080 0.3366 0.3366 0.2080 0.0000<br />
40 t=0.400 0.0000 0.1379 0.2232 0.2232 0.1379 0.0000<br />
45 t=0.450 0.0000 0.0908 0.1470 0.1470 0.0908 0.0000<br />
50 t=0.500 0.0000 0.0595 0.0962 0.0962 0.0595 0.0000
9<br />
Percentage error with r = 1/6, up to t = 1/2<br />
h = 0.200000003, k = 0.00666666729<br />
Using 5 sp<strong>ac</strong>e steps. Need 75 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
7 t=0.047 0.0000 1.6971 2.4336 2.4336 1.6971 0.0000<br />
15 t=0.100 0.0000 0.9163 1.4814 1.4814 0.9163 0.0000<br />
22 t=0.147 0.0000 0.5754 0.9311 0.9311 0.5754 0.0000<br />
30 t=0.200 0.0000 0.3384 0.5475 0.5475 0.3384 0.0000<br />
37 t=0.247 0.0000 0.2126 0.3440 0.3440 0.2126 0.0000<br />
45 t=0.300 0.0000 0.1250 0.2023 0.2023 0.1250 0.0000<br />
52 t=0.347 0.0000 0.0785 0.1271 0.1271 0.0785 0.0000<br />
60 t=0.400 0.0000 0.0462 0.0747 0.0747 0.0462 0.0000<br />
68 t=0.453 0.0000 0.0271 0.0439 0.0439 0.0271 0.0000<br />
75 t=0.500 0.0000 0.0171 0.0276 0.0276 0.0171 0.0000
10<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.100000001, k = 0.00250000018<br />
Using 10 sp<strong>ac</strong>e steps. Need 200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
20 t=0.050 0.0000 0.5751 0.6930 0.6930 0.5751 0.0000<br />
40 t=0.100 0.0000 0.3445 0.5531 0.5531 0.3445 0.0000<br />
60 t=0.150 0.0000 0.2435 0.3939 0.3939 0.2435 0.0000<br />
80 t=0.200 0.0000 0.1696 0.2743 0.2743 0.1696 0.0000<br />
100 t=0.250 0.0000 0.1163 0.1881 0.1881 0.1163 0.0000<br />
120 t=0.300 0.0000 0.0787 0.1274 0.1274 0.0787 0.0000<br />
140 t=0.350 0.0000 0.0528 0.0854 0.0854 0.0528 0.0000<br />
160 t=0.400 0.0000 0.0351 0.0568 0.0568 0.0351 0.0000<br />
180 t=0.450 0.0000 0.0232 0.0375 0.0375 0.0232 0.0000<br />
200 t=0.500 0.0000 0.0152 0.0246 0.0246 0.0152 0.0000
11<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.0500000007, k = 0.000625000044<br />
Using 20 sp<strong>ac</strong>e steps. Need 800 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
80 t=0.050 0.0000 0.1450 0.1723 0.1723 0.1450 0.0000<br />
160 t=0.100 0.0000 0.0862 0.1382 0.1382 0.0862 0.0000<br />
240 t=0.150 0.0000 0.0609 0.0985 0.0985 0.0609 0.0000<br />
320 t=0.200 0.0000 0.0424 0.0687 0.0687 0.0424 0.0000<br />
400 t=0.250 0.0000 0.0291 0.0471 0.0471 0.0291 0.0000<br />
480 t=0.300 0.0000 0.0197 0.0319 0.0319 0.0197 0.0000<br />
560 t=0.350 0.0000 0.0133 0.0214 0.0214 0.0133 0.0000<br />
640 t=0.400 0.0000 0.0088 0.0143 0.0143 0.0088 0.0000<br />
720 t=0.450 0.0000 0.0058 0.0094 0.0094 0.0058 0.0000<br />
800 t=0.500 0.0000 0.0038 0.0062 0.0062 0.0038 0.0000
12<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.0250000004, k = 0.000156250011<br />
Using 40 sp<strong>ac</strong>e steps. Need 3200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
320 t=0.050 0.0000 0.0363 0.0430 0.0430 0.0363 0.0000<br />
640 t=0.100 0.0000 0.0215 0.0345 0.0345 0.0215 0.0000<br />
960 t=0.150 0.0000 0.0152 0.0246 0.0246 0.0152 0.0000<br />
1280 t=0.200 0.0000 0.0106 0.0172 0.0172 0.0106 0.0000<br />
1600 t=0.250 0.0000 0.0073 0.0118 0.0118 0.0073 0.0000<br />
1920 t=0.300 0.0000 0.0049 0.0080 0.0080 0.0049 0.0000<br />
2240 t=0.350 0.0000 0.0033 0.0053 0.0053 0.0033 0.0000<br />
2560 t=0.400 0.0000 0.0022 0.0036 0.0036 0.0022 0.0000<br />
2880 t=0.450 0.0000 0.0015 0.0024 0.0024 0.0015 0.0000<br />
3200 t=0.500 0.0000 0.0010 0.0016 0.0016 0.0010 0.0000
13<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.0125000002, k = 3.90625028E-05<br />
Using 80 sp<strong>ac</strong>e steps. Need 12800 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
1280 t=0.050 0.0000 0.0091 0.0108 0.0107 0.0091 0.0000<br />
2560 t=0.100 0.0000 0.0054 0.0087 0.0086 0.0054 0.0000<br />
3840 t=0.150 0.0000 0.0039 0.0062 0.0062 0.0039 0.0000<br />
5120 t=0.200 0.0000 0.0028 0.0046 0.0046 0.0028 0.0000<br />
6400 t=0.250 0.0000 0.0019 0.0032 0.0032 0.0019 0.0000<br />
7680 t=0.300 0.0000 0.0012 0.0020 0.0020 0.0012 0.0000<br />
8960 t=0.350 0.0000 0.0006 0.0011 0.0011 0.0006 0.0000<br />
10240 t=0.400 0.0000 0.0003 0.0005 0.0005 0.0003 0.0000<br />
11520 t=0.450 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
12800 t=0.500 0.0000 0.0002 0.0002 0.0002 0.0002 0.0000
14<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.00625000009, k = 9.76562569E-06<br />
Using 160 sp<strong>ac</strong>e steps. Need 51200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
5120 t=0.050 0.0000 0.0022 0.0027 0.0027 0.0022 0.0000<br />
10240 t=0.100 0.0000 0.0007 0.0015 0.0015 0.0007 0.0000<br />
15360 t=0.150 0.0000 0.0003 0.0002 0.0002 0.0003 0.0000<br />
20480 t=0.200 0.0000 0.0009 0.0013 0.0013 0.0009 0.0000<br />
25600 t=0.250 0.0000 0.0013 0.0019 0.0019 0.0013 0.0000<br />
30720 t=0.300 0.0000 0.0017 0.0024 0.0024 0.0017 0.0000<br />
35840 t=0.350 0.0000 0.0017 0.0026 0.0026 0.0017 0.0000<br />
40960 t=0.400 0.0000 0.0017 0.0027 0.0027 0.0017 0.0000<br />
46080 t=0.450 0.0000 0.0017 0.0028 0.0028 0.0017 0.0000<br />
51200 t=0.500 0.0000 0.0017 0.0029 0.0029 0.0017 0.0000
15<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.00312500005, k = 2.44140642E-06<br />
Using 320 sp<strong>ac</strong>e steps. Need 204800 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
20480 t=0.050 0.0000 0.0008 0.0002 0.0002 0.0008 0.0000<br />
40960 t=0.100 0.0000 0.0030 0.0036 0.0036 0.0030 0.0000<br />
61440 t=0.150 0.0000 0.0043 0.0064 0.0064 0.0043 0.0000<br />
81920 t=0.200 0.0000 0.0050 0.0079 0.0079 0.0050 0.0000<br />
102400 t=0.250 0.0000 0.0053 0.0088 0.0088 0.0053 0.0000<br />
122880 t=0.300 0.0000 0.0055 0.0094 0.0094 0.0055 0.0000<br />
143360 t=0.350 0.0000 0.0058 0.0103 0.0103 0.0058 0.0000<br />
163840 t=0.400 0.0000 0.0062 0.0119 0.0119 0.0062 0.0000<br />
184320 t=0.450 0.0000 0.0068 0.0139 0.0139 0.0068 0.0000<br />
204800 t=0.500 0.0000 0.0085 0.0158 0.0158 0.0085 0.0000
16<br />
Percentage error with r = 1/4, up to t = 1/2<br />
h = 0.00156250002, k = 6.10351606E-07<br />
Using 640 sp<strong>ac</strong>e steps. Need 819200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
81920 t=0.050 0.0000 0.0038 0.0028 0.0028 0.0038 0.0000<br />
163840 t=0.100 0.0000 0.0068 0.0110 0.0110 0.0068 0.0000<br />
245760 t=0.150 0.0000 0.0119 0.0223 0.0223 0.0119 0.0000<br />
327680 t=0.200 0.0000 0.0161 0.0312 0.0312 0.0161 0.0000<br />
409600 t=0.250 0.0000 0.0199 0.0382 0.0382 0.0199 0.0000<br />
491520 t=0.300 0.0000 0.0236 0.0454 0.0454 0.0236 0.0000<br />
573440 t=0.350 0.0000 0.0296 0.0641 0.0641 0.0296 0.0000<br />
655360 t=0.400 0.0000 0.0319 0.0697 0.0697 0.0319 0.0000<br />
737280 t=0.450 0.0000 0.0421 0.0635 0.0635 0.0421 0.0000<br />
819200 t=0.500 0.0000 0.0441 0.0741 0.0741 0.0441 0.0000
17<br />
Solving the <strong>Parabolic</strong> PDE Using the Crank-Nicolson Method<br />
We will use the scheme<br />
−rφ i−1,j+1 + (2 + 2r)φ i,j+1 − rφ i+1,j+1 = rφ i−1,j + (2 − 2r)φ i,j + rφ i+1,j<br />
where r = k/h 2 , with x = hi <strong>and</strong> t = jk.
18<br />
Solution with r = 1: first few steps<br />
h = 0.200000003, k = 0.0400000028<br />
Using 5 sp<strong>ac</strong>e steps. Need 12 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
1 t=0.040 1.0000 0.5455 0.1818 0.1818 0.5455 1.0000<br />
2 t=0.080 1.0000 0.6612 0.4628 0.4628 0.6612 1.0000<br />
3 t=0.120 1.0000 0.7739 0.6326 0.6326 0.7739 1.0000<br />
4 t=0.160 1.0000 0.8458 0.7508 0.7508 0.8458 1.0000
19<br />
Solution with r = 1, up to t = 1/2<br />
h = 0.100000001, k = 0.0100000007<br />
Using 10 sp<strong>ac</strong>e steps. Need 50 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000<br />
5 t=0.050 1.0000 0.5404 0.2670 0.2670 0.5404 1.0000<br />
10 t=0.100 1.0000 0.7213 0.5492 0.5492 0.7213 1.0000<br />
15 t=0.150 1.0000 0.8293 0.7237 0.7237 0.8293 1.0000<br />
20 t=0.200 1.0000 0.8954 0.8307 0.8307 0.8954 1.0000<br />
25 t=0.250 1.0000 0.9359 0.8963 0.8963 0.9359 1.0000<br />
30 t=0.300 1.0000 0.9607 0.9364 0.9364 0.9607 1.0000<br />
35 t=0.350 1.0000 0.9759 0.9611 0.9611 0.9759 1.0000<br />
40 t=0.400 1.0000 0.9853 0.9761 0.9761 0.9853 1.0000<br />
45 t=0.450 1.0000 0.9910 0.9854 0.9854 0.9910 1.0000<br />
50 t=0.500 1.0000 0.9945 0.9910 0.9910 0.9945 1.0000
20<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.100000001, k = 0.0100000007<br />
Using 10 sp<strong>ac</strong>e steps. Need 50 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
5 t=0.050 0.0000 0.2074 0.3281 0.3281 0.2074 0.0000<br />
10 t=0.100 0.0000 0.0267 0.0437 0.0437 0.0267 0.0000<br />
15 t=0.150 0.0000 0.0459 0.0742 0.0742 0.0459 0.0000<br />
20 t=0.200 0.0000 0.0662 0.1071 0.1071 0.0662 0.0000<br />
25 t=0.250 0.0000 0.0638 0.1032 0.1032 0.0638 0.0000<br />
30 t=0.300 0.0000 0.0533 0.0862 0.0862 0.0533 0.0000<br />
35 t=0.350 0.0000 0.0413 0.0668 0.0668 0.0413 0.0000<br />
40 t=0.400 0.0000 0.0306 0.0495 0.0495 0.0306 0.0000<br />
45 t=0.450 0.0000 0.0220 0.0356 0.0356 0.0220 0.0000<br />
50 t=0.500 0.0000 0.0154 0.0250 0.0250 0.0154 0.0000
21<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.0500000007, k = 0.00250000018<br />
Using 20 sp<strong>ac</strong>e steps. Need 200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
20 t=0.050 0.0000 0.0268 0.0927 0.0927 0.0267 0.0000<br />
40 t=0.100 0.0000 0.0016 0.0040 0.0040 0.0016 0.0000<br />
60 t=0.150 0.0000 0.0155 0.0251 0.0251 0.0155 0.0000<br />
80 t=0.200 0.0000 0.0198 0.0320 0.0320 0.0198 0.0000<br />
100 t=0.250 0.0000 0.0184 0.0298 0.0298 0.0184 0.0000<br />
120 t=0.300 0.0000 0.0151 0.0245 0.0245 0.0152 0.0000<br />
140 t=0.350 0.0000 0.0116 0.0188 0.0188 0.0116 0.0000<br />
160 t=0.400 0.0000 0.0086 0.0139 0.0139 0.0086 0.0000<br />
180 t=0.450 0.0000 0.0061 0.0099 0.0099 0.0062 0.0000<br />
200 t=0.500 0.0000 0.0043 0.0070 0.0070 0.0043 0.0000
22<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.0250000004, k = 0.000625000044<br />
Using 40 sp<strong>ac</strong>e steps. Need 800 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
80 t=0.050 0.0000 0.0049 0.0237 0.0237 0.0048 0.0000<br />
160 t=0.100 0.0000 0.0000 0.0004 0.0004 0.0001 0.0000<br />
240 t=0.150 0.0000 0.0043 0.0070 0.0070 0.0044 0.0000<br />
320 t=0.200 0.0000 0.0054 0.0088 0.0088 0.0055 0.0000<br />
400 t=0.250 0.0000 0.0051 0.0082 0.0082 0.0051 0.0000<br />
480 t=0.300 0.0000 0.0043 0.0069 0.0069 0.0043 0.0000<br />
560 t=0.350 0.0000 0.0034 0.0054 0.0054 0.0034 0.0000<br />
640 t=0.400 0.0000 0.0026 0.0042 0.0042 0.0026 0.0000<br />
720 t=0.450 0.0000 0.0020 0.0032 0.0032 0.0020 0.0000<br />
800 t=0.500 0.0000 0.0015 0.0024 0.0025 0.0016 0.0000
23<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.0125000002, k = 0.000156250011<br />
Using 80 sp<strong>ac</strong>e steps. Need 3200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
320 t=0.050 0.0000 0.0009 0.0058 0.0057 0.0008 0.0000<br />
640 t=0.100 0.0000 0.0006 0.0006 0.0007 0.0006 0.0000<br />
960 t=0.150 0.0000 0.0020 0.0030 0.0030 0.0020 0.0000<br />
1280 t=0.200 0.0000 0.0025 0.0039 0.0039 0.0026 0.0000<br />
1600 t=0.250 0.0000 0.0026 0.0041 0.0041 0.0027 0.0000<br />
1920 t=0.300 0.0000 0.0026 0.0040 0.0040 0.0027 0.0000<br />
2240 t=0.350 0.0000 0.0025 0.0039 0.0039 0.0026 0.0000<br />
2560 t=0.400 0.0000 0.0024 0.0037 0.0037 0.0025 0.0000<br />
2880 t=0.450 0.0000 0.0023 0.0035 0.0036 0.0023 0.0000<br />
3200 t=0.500 0.0000 0.0022 0.0034 0.0034 0.0023 0.0000
24<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.00625000009, k = 3.90625028E-05<br />
Using 160 sp<strong>ac</strong>e steps. Need 12800 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
1280 t=0.050 0.0000 0.0008 0.0006 0.0006 0.0009 0.0000<br />
2560 t=0.100 0.0000 0.0025 0.0031 0.0031 0.0026 0.0000<br />
3840 t=0.150 0.0000 0.0041 0.0058 0.0058 0.0042 0.0000<br />
5120 t=0.200 0.0000 0.0054 0.0078 0.0078 0.0054 0.0000<br />
6400 t=0.250 0.0000 0.0064 0.0093 0.0093 0.0064 0.0000<br />
7680 t=0.300 0.0000 0.0070 0.0103 0.0104 0.0071 0.0000<br />
8960 t=0.350 0.0000 0.0074 0.0111 0.0111 0.0074 0.0000<br />
10240 t=0.400 0.0000 0.0077 0.0116 0.0116 0.0078 0.0000<br />
11520 t=0.450 0.0000 0.0079 0.0119 0.0119 0.0080 0.0000<br />
12800 t=0.500 0.0000 0.0081 0.0121 0.0121 0.0081 0.0000
25<br />
Percentage error with r = 1, up to t = 1/2<br />
h = 0.00312500005, k = 9.76562569E-06<br />
Using 320 sp<strong>ac</strong>e steps. Need 51200 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
5120 t=0.050 0.0000 0.0043 0.0031 0.0031 0.0044 0.0000<br />
10240 t=0.100 0.0000 0.0100 0.0122 0.0123 0.0100 0.0000<br />
15360 t=0.150 0.0000 0.0156 0.0214 0.0215 0.0157 0.0000<br />
20480 t=0.200 0.0000 0.0206 0.0293 0.0293 0.0207 0.0000<br />
25600 t=0.250 0.0000 0.0242 0.0354 0.0354 0.0242 0.0000<br />
30720 t=0.300 0.0000 0.0271 0.0399 0.0400 0.0271 0.0000<br />
35840 t=0.350 0.0000 0.0292 0.0432 0.0433 0.0293 0.0000<br />
40960 t=0.400 0.0000 0.0306 0.0456 0.0457 0.0307 0.0000<br />
46080 t=0.450 0.0000 0.0315 0.0472 0.0473 0.0317 0.0000<br />
51200 t=0.500 0.0000 0.0322 0.0481 0.0482 0.0323 0.0000
26<br />
Percentage error with r = 10, up to t = 1/2<br />
h = 0.0125000002, k = 0.00156250002<br />
Using 80 sp<strong>ac</strong>e steps. Need 320 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
32 t=0.050 0.0000 0.0049 0.0046 0.0046 0.0049 0.0000<br />
64 t=0.100 0.0000 0.0006 0.0009 0.0009 0.0006 0.0000<br />
96 t=0.150 0.0000 0.0006 0.0010 0.0010 0.0006 0.0000<br />
128 t=0.200 0.0000 0.0010 0.0016 0.0016 0.0010 0.0000<br />
160 t=0.250 0.0000 0.0010 0.0016 0.0016 0.0010 0.0000<br />
192 t=0.300 0.0000 0.0008 0.0014 0.0014 0.0009 0.0000<br />
224 t=0.350 0.0000 0.0007 0.0011 0.0011 0.0007 0.0000<br />
256 t=0.400 0.0000 0.0005 0.0009 0.0009 0.0006 0.0000<br />
288 t=0.450 0.0000 0.0004 0.0007 0.0007 0.0004 0.0000<br />
320 t=0.500 0.0000 0.0003 0.0005 0.0006 0.0004 0.0000
27<br />
Percentage error with r = 40, up to t = 1/2<br />
h = 0.0125000002, k = 0.00625000009<br />
Using 80 sp<strong>ac</strong>e steps. Need 80 time steps.<br />
x=0.00 x=0.20 x=0.40 x=0.60 x=0.80 x=1.00<br />
0 t=0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000<br />
8 t=0.050 0.0000 0.1524 0.0148 0.0148 0.1523 0.0000<br />
16 t=0.100 0.0000 0.0075 0.0130 0.0129 0.0073 0.0000<br />
24 t=0.150 0.0000 0.0004 0.0109 0.0108 0.0006 0.0000<br />
32 t=0.200 0.0000 0.0053 0.0080 0.0079 0.0050 0.0000<br />
40 t=0.250 0.0000 0.0048 0.0056 0.0054 0.0045 0.0000<br />
48 t=0.300 0.0000 0.0029 0.0037 0.0035 0.0026 0.0000<br />
56 t=0.350 0.0000 0.0015 0.0023 0.0022 0.0012 0.0000<br />
64 t=0.400 0.0000 0.0008 0.0014 0.0012 0.0005 0.0000<br />
72 t=0.450 0.0000 0.0004 0.0007 0.0005 0.0001 0.0000<br />
80 t=0.500 0.0000 0.0001 0.0002 0.0000 0.0001 0.0000
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