Lab 05 Transient Heat Conduction with X-Dependent Source

Problem Statement: Using the Lab05 matlab script as a starting point, fill in the details for transient heat conduction with an x-dependent source of s(x)=0.1*sin((x/L)*pi). For the 32 element grid and Theta=0.5, start with a time step size of 100 seconds and uniformly refine it until the midpoint solution to converges to 1e-4 at the final time of t_f=2000 seconds. Repeat for both Theta=0 and Theta=1.0.
Then present a plot of the t_fsolution for each value of Theta for a time step of 12.5 seconds, commenting on the accuracy of each solution.
Finally, report and discuss temporal convergence rates in Enorm at the final time for each value of Theta.

Solution: Using the Transient Heat Conduction handout as a reference, the MatLab code was generated to solve the problem statement.

After running the code for Theta = 0 (forward TS / Explicit Euler) starting at a time step of 100, I found out that the solution diverged and did not solve. The magnitudes for the temperature were in the order of E26. So I didn’t get any good results for theta = 0

After running the code for Theta = .5 (backward TS / Implicit Euler) the following spreadsheet was recorded.

*Convergence Table For Theta = .5*
Distance	Time Step 100	Time Step 50	Time Step 25	Time Step 12.5	Time Step 6.25
0	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000
0.03125	273.001806228521	273.001755400060	273.001730578203	273.001718311785	273.001712214268
0.0625	273.003595062074	273.003493894660	273.003444489993	273.003420075289	273.003407938977
0.09375	273.005349273215	273.005198741143	273.005125229460	273.005088901598	273.005070843370
0.125	273.007051967930	273.006853520907	273.006756610167	273.006708719003	273.006684912770
0.15625	273.008686748339	273.008442297519	273.008322921024	273.008263927776	273.008234602806
0.1875	273.010237870612	273.009949770190	273.009809077602	273.009739550407	273.009704989115
0.21875	273.011690396596	273.011361421135	273.011200767400	273.011121375844	273.011081911073
0.25	273.013030337669	273.012663655384	273.012484587685	273.012396096352	273.012352108170
0.28125	273.014244789467	273.013843931709	273.013648174566	273.013551435675	273.013503347713
0.3125	273.015322056156	273.014890883404	273.014680322066	273.014576267265	273.014524542636
0.34375	273.016251763068	273.015794427751	273.015571090038	273.015460721433	273.015405858273
0.375	273.017024956618	273.016545863125	273.016311899902	273.016196280402	273.016138807073
0.40625	273.017634190529	273.017137952792	273.016895617252	273.016775860337	273.016716330339
0.4375	273.018073597545	273.017564994605	273.017316620572	273.017193879566	273.017132866206
0.46875	273.018338945936	273.017822875919	273.017570855370	273.017446312336	273.017384403205
0.5	273.018427680254	273.017909113194	273.017655873225	273.017530727581	273.017468518898
0.53125	273.018338945936	273.017822875919	273.017570855370	273.017446312336	273.017384403205
0.5625	273.018073597545	273.017564994605	273.017316620572	273.017193879566	273.017132866206
0.59375	273.017634190529	273.017137952792	273.016895617252	273.016775860337	273.016716330339
0.625	273.017024956618	273.016545863125	273.016311899902	273.016196280402	273.016138807073
0.65625	273.016251763068	273.015794427751	273.015571090038	273.015460721433	273.015405858273
0.6875	273.015322056156	273.014890883404	273.014680322066	273.014576267265	273.014524542636
0.71875	273.014244789467	273.013843931709	273.013648174566	273.013551435675	273.013503347713
0.75	273.013030337669	273.012663655384	273.012484587685	273.012396096352	273.012352108170
0.78125	273.011690396596	273.011361421135	273.011200767400	273.011121375844	273.011081911073
0.8125	273.010237870612	273.009949770190	273.009809077602	273.009739550407	273.009704989115
0.84375	273.008686748339	273.008442297519	273.008322921024	273.008263927776	273.008234602806
0.875	273.007051967930	273.006853520907	273.006756610167	273.006708719003	273.006684912770
0.90625	273.005349273215	273.005198741143	273.005125229460	273.005088901598	273.005070843370
0.9375	273.003595062074	273.003493894660	273.003444489993	273.003420075289	273.003407938977
0.96875	273.001806228521	273.001755400060	273.001730578203	273.001718311785	273.001712214268
1	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000

	Change in T @ x = .5	0.000518567060	0.000253239969	0.000125145644	0.000062208683

After running the code for Theta = 1 (Trapezoid Rule / Crank - Nicholson) the following spreadsheet was recorded.

*Convergence Table For Theta =1*
	Time Step 100	Time Step 50	Time Step 25	Time Step 12.5	Time Step 6.25	Time Step 3.125
0	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000
0.03125	273.001888913675	273.001795617964	273.001750409661	273.001728158603	273.001717120507	273.001711623213
0.0625	273.003759636080	273.003573943145	273.003483961921	273.003439674095	273.003417704205	273.003406762559
0.09375	273.005594151134	273.005317849301	273.005183961723	273.005118063644	273.005085373544	273.005069092919
0.125	273.007374791454	273.007010541660	273.006834037140	273.006747163443	273.006704067955	273.006682605144
0.15625	273.009084408501	273.008635718672	273.008418297043	273.008311284370	273.008258198527	273.008231760228
0.1875	273.010706537729	273.010177728991	273.009921484144	273.009795363085	273.009732798133	273.009701638961
0.21875	273.012225557145	273.011621722217	273.011329121929	273.011185107098	273.011113665573	273.011078085607
0.25	273.013626837763	273.012953791904	273.012627654076	273.012467132416	273.012387502337	273.012347844233
0.28125	273.014896884483	273.014161109495	273.013804575012	273.013629092433	273.013542040683	273.013498686369
0.3125	273.016023466063	273.015232047859	273.014848550344	273.014659796840	273.014566161773	273.014519528776
0.34375	273.016995732906	273.016156293276	273.015749526020	273.015549319389	273.015450002764	273.015400540184
0.375	273.017804321549	273.016924944757	273.016498825150	273.016289093493	273.016185051781	273.016133235971
0.40625	273.018441444843	273.017530599768	273.017089231575	273.016871994721	273.016764229903	273.016710559876
0.4375	273.018900966943	273.017967425519	273.017515059357	273.017292409418	273.017181959326	273.017126951955
0.46875	273.019178462398	273.018231215141	273.017772207544	273.017546288758	273.017434217088	273.017378402124
0.5	273.019271258776	273.018319428195	273.017858199656	273.017631187746	273.017518573810	273.017462488780
0.53125	273.019178462398	273.018231215141	273.017772207544	273.017546288758	273.017434217088	273.017378402124
0.5625	273.018900966943	273.017967425519	273.017515059357	273.017292409418	273.017181959326	273.017126951955
0.59375	273.018441444843	273.017530599768	273.017089231575	273.016871994721	273.016764229903	273.016710559876
0.625	273.017804321549	273.016924944757	273.016498825150	273.016289093493	273.016185051781	273.016133235971
0.65625	273.016995732906	273.016156293276	273.015749526020	273.015549319389	273.015450002764	273.015400540184
0.6875	273.016023466063	273.015232047859	273.014848550344	273.014659796840	273.014566161773	273.014519528776
0.71875	273.014896884483	273.014161109495	273.013804575012	273.013629092433	273.013542040683	273.013498686369
0.75	273.013626837763	273.012953791904	273.012627654076	273.012467132416	273.012387502337	273.012347844233
0.78125	273.012225557145	273.011621722217	273.011329121929	273.011185107098	273.011113665573	273.011078085607
0.8125	273.010706537729	273.010177728991	273.009921484144	273.009795363085	273.009732798133	273.009701638961
0.84375	273.009084408501	273.008635718672	273.008418297043	273.008311284370	273.008258198527	273.008231760228
0.875	273.007374791454	273.007010541660	273.006834037140	273.006747163443	273.006704067955	273.006682605144
0.90625	273.005594151134	273.005317849301	273.005183961723	273.005118063644	273.005085373544	273.005069092919
0.9375	273.003759636080	273.003573943145	273.003483961921	273.003439674095	273.003417704205	273.003406762559
0.96875	273.001888913675	273.001795617964	273.001750409661	273.001728158603	273.001717120507	273.001711623213
1	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000	273.000000000000

	Change in T @ x = .5	0.000951830581	0.000461228539	0.000227011910	0.000112613936	0.000056085030

For Theta = 0, which is the Implicit Euler, the code had to be changed. The max iterations was set to 1 and the time step had to be reduced really really small before it would converge. It finally converged at 3.125. The following table was recorded.

*Convergence Table For Theta = 0*
Distance	Time Step 100	Time Step 50	Time Step 25	Time Step 12.5	Time Step 6.25	Time Step 3.125	Time Step 1.5625
0	273.00000	2.7300E+02	2.7300E+02	2.7300E+02	273.0000000	273.0000000	273.0000000
0.03125	-6198398394.20106	-1.2022E+23	-3.9058E+34	-4.0965E+20	273.0069157	273.0069126	273.0069110
0.0625	12341295187.67570	2.3246E+23	7.7103E+34	8.1535E+20	273.0137648	273.0137586	273.0137555
0.09375	-18276259565.34410	-3.2931E+23	-1.1319E+35	-1.2132E+21	273.0204813	273.0204721	273.0204675
0.125	23689031515.28790	4.0444E+23	1.4649E+35	1.5994E+21	273.0270006	273.0269885	273.0269824
0.15625	-28094620911.05710	-4.5314E+23	-1.7636E+35	-1.9701E+21	273.0332599	273.0332449	273.0332374
0.1875	30894016750.94040	4.7266E+23	2.0234E+35	2.3219E+21	273.0391988	273.0391812	273.0391723
0.21875	-31488395907.33360	-4.6254E+23	-2.2423E+35	-2.6513E+21	273.0447603	273.0447401	273.0447300
0.25	29426149966.82190	4.2474E+23	2.4203E+35	2.9552E+21	273.0498907	273.0498682	273.0498569
0.28125	-24546813242.01570	-3.6353E+23	-2.5595E+35	-3.2307E+21	273.0545406	273.0545160	273.0545037
0.3125	17083750199.97740	2.8524E+23	2.6636E+35	3.4750E+21	273.0586652	273.0586388	273.0586255
0.34375	-7695079322.53958	-1.9772E+23	-2.7379E+35	-3.6858E+21	273.0622249	273.0621968	273.0621828
0.375	-2591699479.52627	1.0970E+23	2.7879E+35	3.8612E+21	273.0651853	273.0651559	273.0651412
0.40625	12516002791.10110	-2.9971E+22	-2.8195E+35	-3.9993E+21	273.0675179	273.0674875	273.0674723
0.4375	-20777166320.20460	-3.3500E+22	2.8378E+35	4.0990E+21	273.0692003	273.0691691	273.0691535
0.46875	26247038577.11340	7.4335E+22	-2.8470E+35	-4.1592E+21	273.0702163	273.0701846	273.0701688
0.5	-28160536780.63160	-8.8421E+22	2.8497E+35	4.1793E+21	273.0705561	273.0705242	273.0705083
0.53125	26247038577.11340	7.4335E+22	-2.8470E+35	-4.1592E+21	273.0702163	273.0701846	273.0701688
0.5625	-20777166320.20460	-3.3500E+22	2.8378E+35	4.0990E+21	273.0692003	273.0691691	273.0691535
0.59375	12516002791.10110	-2.9971E+22	-2.8195E+35	-3.9993E+21	273.0675179	273.0674875	273.0674723
0.625	-2591699479.52627	1.0970E+23	2.7879E+35	3.8612E+21	273.0651853	273.0651559	273.0651412
0.65625	-7695079322.53958	-1.9772E+23	-2.7379E+35	-3.6858E+21	273.0622249	273.0621968	273.0621828
0.6875	17083750199.97740	2.8524E+23	2.6636E+35	3.4750E+21	273.0586652	273.0586388	273.0586255
0.71875	-24546813242.01570	-3.6353E+23	-2.5595E+35	-3.2307E+21	273.0545406	273.0545160	273.0545037
0.75	29426149966.82200	4.2474E+23	2.4203E+35	2.9552E+21	273.0498907	273.0498682	273.0498569
0.78125	-31488395907.33370	-4.6254E+23	-2.2423E+35	-2.6513E+21	273.0447603	273.0447401	273.0447300
0.8125	30894016750.94050	4.7266E+23	2.0234E+35	2.3219E+21	273.0391988	273.0391812	273.0391723
0.84375	-28094620911.05710	-4.5314E+23	-1.7636E+35	-1.9701E+21	273.0332599	273.0332449	273.0332374
0.875	23689031515.28790	4.0444E+23	1.4649E+35	1.5994E+21	273.0270006	273.0269885	273.0269824
0.90625	-18276259565.34410	-3.2931E+23	-1.1319E+35	-1.2132E+21	273.0204813	273.0204721	273.0204675
0.9375	12341295187.67570	2.3246E+23	7.7103E+34	8.1535E+20	273.0137648	273.0137586	273.0137555
0.96875	-6198398394.20107	-1.2022E+23	-3.9058E+34	-4.0965E+20	273.0069157	273.0069126	273.0069110
1	273.00000	2.7300E+02	2.7300E+02	2.7300E+02	273.0000000	273.0000000	273.0000000

	Change in T @ x = .5	8.8421E+22	-2.8497E+35	2.8497E+35	4.1793E+21	0.0000318	0.0000159

To see a plot of the t_fsolution for each value of Theta for a time step of 12.5 click here. Note: Since the Theta = 0 solution didn’t converge at T.S 12.5 it wasn’t added to the graph.

The temporal convergence rates for all three methods are tabulated in the tables below. The slope for the temporal convergence is related to

the equation C_t Dt^f(o). Where f(Q) is 1 for Implicit and Explicit Euler and 2 for Crank Nicholson. So the slope for Implicit and Explicit Euler should be equal to one in the adequate mesh region and Crank Nicholson should be equal to 2 in the adequate mesh region. One can see this in the results below.

Temporal Convergence Rates For Theta = 0, .5, 1



Explicit Euler Theta = 0 Convergence Table
	Time Step	Q^h	e^h/2	m^h/4
t^h	100	-2.816054E+10
t^h/2	50	-8.8421E+22	-8.8421E+22
t^h/4	25	2.8497E+35	2.8497E+35	#NUM!
t^h/8	12.5	4.1793E+21	-2.8497E+35	#NUM!
t^h/16	6.25	273.0705561	-4.1793E+21	4.5955E+01
t^h/32	3.125	273.0705242	-3.1826E-05	8.6763E+01
t^h/64	1.5625	273.0705083	-1.5902E-05	1.0010E+00


Crank - Nicholson Theta = .5 Convergence Table
		Q^h	e^h/2	m^h/4
t^h	100	273.01842768
t^h/2	50	273.01790911	-0.00051857
t^h/4	25	273.01765587	-0.00025324	2.068050887
t^h/8	12.5	273.01753073	-0.00012515	2.033794097
t^h/16	6.25	273.01746852	-0.00006221	2.016840413

Implicit Euler Theta = 1 Convergence Table
		Q^h	e^h/2	m^h/4
t^h	100	273.019271258776
t^h/2	50	273.018319428195	-0.000951830581
t^h/4	25	273.017858199656	-0.000461228539	1.045223023
t^h/8	12.5	273.017631187746	-0.000227011910	1.022713795
t^h/16	6.25	273.017518573810	-0.000112613936	1.011382617
t^h/32	3.125	273.017462488780	-0.000056085030	1.005697724

Conclusion: From the temporal convergence results you can see that the Explicit and Implicit Euler had a slope of 1 where as the Crank – Nicholson had a slope of 2. The Explicit Euler didn’t require any iteration, which is good but it took a very very small time step for it to converge. From the comparison graph at a time step of 12.5 both the Crank Nicholson and Implicit Euler converged very nicely. They converged to almost exactly the same answer but depending on if you want time accurate results or steady state results their accuracies are different. If you want a time accurate solution Crank – Nicholson will give you the most time accurate results where as the Implicit Euler will give you the most accurate Steady State Results.