Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method
Aim:
to derive the following 4th order approximations of the second-order derivative.
1. Central difference
2. Skewed right-sided difference
3. Skewed left-sided difference.
given:
After deriving these schemes, prove that your skewed schemes are fourth-order accurate i.e the thumb rule given in HINT (ii).
Once you have done this, write a program in Matlab to evaluate the second-order derivative of the analytical function exp(x)*cos(x) and compare it with the 3 numerical approximations that you have derived.
Provide a plot that compares the absolute error between the above-mentioned schemes
In your own words, describe why a skewed scheme is useful? What can a skewed scheme do that a CD scheme cannot?
HINT :
i) The order of accuracy P = Number of nodes in the stencil N - order of derivative Q + 1 for symmetric/central schemes
ii) The order of accuracy P = Number of nodes in the stencil N - order of derivative Q for one-sided/skewed schemes
iii) Solve the system of equations using the matrix inversion method in Matlab to obtain the taylor table coefficients and use them directly in the numerical approximation statements to avoid input errors.
Taylor's Series:
A Taylor Series is an expansion of some function into an infinite sum of terms, where each term has a larger exponent like x, x2, x3, etc.
Example: The taylor series for the function ex��:
ex = 1+x+ x22! + x33! + x44! + x55! +...
says that the function:ex is equal to the infinite sum of terms:1 + x + x2/2! + x3/3! + ... etc.
derivation:
Coding:
lets take or assume f(x) = exp(x) * cos(x)
also f''(x)=-2 exp(x) * sin(x)........analytical derivative function equation
clear all
close all
clc
%f(x) = exp. (x) * cos(x)
%f''(x) = -2 exp(x)*sin(x)
x = pi/3;
dx = linspace(pi/4,pi/400,100);
% calculating analytical value of second order function
analytical_derivative = -2*exp(x)*sin(x);
%central differencing scheme
a_central = [1 1 1 1 1;-2 -1 0 1 2;2 1/2 0 1/2 2;-4/3 -1/6 0 1/6 4/3;2/3 1/24 0 1/24 2/3];
b_central = [0;0;1;0;0]
c_central = inv(a_central)*b_central;
% c_central(1)*(exp(x-(dx)*2).*cos(x-(dx)*2))
%calculative second order derivative
central_differencing = (c_central(1)*(exp(x-(dx)*2).*cos(x-(dx)*2)))+(c_central(2)*(exp(x-(dx)).*cos(x-(dx))))+(c_central(1)*(exp(x).*cos(x)))+(c_central(1)*(exp(x+(dx)).*cos(x+(dx))))+(c_central(1)*(exp(x+(dx)*2).*cos(x+(dx)*2)))
% error
error_central = abs(analytical_derivative-central_differencing)
% skewed right sided differnce
a_right = [1 1 1 1 1 1;0 1 2 3 4 5;0 1/2 4/2 9/2 16/2 25/2;0 1/6 8/6 27/6 64/6 125/6;0 1/24 16/24 81/24 256/24 625/24;0 1/120 32/120 243/120 1024/120 3125/120];
b_right = [0;0;1;0;0;0]
c_right = inv(a_right)*b_right;
%calaculating right sided skewed difference c_right(1)*(exp(x-(dx)*2).*cos(x-(dx)*2))
right_differencing = (c_right(1)*(exp(x).*cos(x)))+(c_right(2)*(exp(x+(dx)).*cos(x+(dx))))+(c_right(3)*(exp(x+(dx)*2).*cos(x+(dx)*2)))+(c_right(4)*(exp(x+(dx)*3).*cos(x+(dx)*3)))+(c_right(5)*(exp(x+(dx)*4).*cos(x+(dx)*4)))+(c_right(6)*(exp(x+(dx)*5).*cos(x+(dx)*5)))
% error
error_right = abs(analytical_derivative-right_differencing)
% skewed left sided difference
a_left = [1 1 1 1 1 1;-5 -4 -3 -2 -1 0;25/2 16/2 9/2 4/2 1/2 0;-125/6 -64/6 -27/6 -8/6 -1/6 0;625/24 256/24 81/24 16/24 1/24 0;-3125/120 -1024/120 -243/120 -32/120 -1/120 0]
b_left = [0;0;1;0;0;0]
c_left = inv(a_left)*b_left;
%calculating left sided differnce c_left(1)*(exp(x).*cos(x))
left_differncing = (c_left(1)*(exp(x-(dx)*5).*cos(x-(dx)*5)))+(c_left(2)*(exp(x-(dx)*4).*cos(x-(dx)*4)))+(c_left(3)*(exp(x-(dx)*3).*cos(x-(dx)*3)))+(c_left(4)*(exp(x-(dx)*2).*cos(x-(dx)*2)))+(c_left(5)*(exp(x-(dx)).*cos(x-(dx))))+(c_left(6)*(exp(x).*cos(x)));
%error
error_left = abs(analytical_derivative-left_differncing);
% plotting
figure(1)
plot(dx,error_central,'b','LineWidth',2)
hold on
grid on
plot(dx,error_right,'g','LineWidth',2)
hold on
plot(dx,error_left,'r','LineWidth',2)
xlabel("dx")
ylabel("error")
legend("central differncing","right differencing","left differncing");
figure(2)
loglog(dx,error_central,'b','LineWidth',2)
hold on
grid on
loglog(dx,error_right,'g','LineWidth',2)
hold on
loglog(dx,error_left,'r','LineWidth',2)
xlabel("dx")
ylabel("error")
legend("central differncing","right differencing","left differncing");OUTPUT :
central_differencing =
Columns 1 through 7
2.6359 2.6151 2.5946 2.5743 2.5543 2.5346 2.5152
Columns 8 through 14
2.4960 2.4771 2.4585 2.4402 2.4222 2.4044 2.3869
Columns 15 through 21
2.3696 2.3527 2.3360 2.3195 2.3033 2.2874 2.2718
Columns 22 through 28
2.2563 2.2412 2.2262 2.2116 2.1971 2.1829 2.1689
Columns 29 through 35
2.1552 2.1417 2.1284 2.1153 2.1024 2.0897 2.0772
Columns 36 through 42
2.0650 2.0529 2.0410 2.0293 2.0177 2.0063 1.9951
Columns 43 through 49
1.9841 1.9732 1.9625 1.9519 1.9415 1.9312 1.9210
Columns 50 through 56
1.9109 1.9010 1.8912 1.8814 1.8718 1.8623 1.8529
Columns 57 through 63
1.8435 1.8342 1.8250 1.8159 1.8068 1.7978 1.7888
Columns 64 through 70
1.7799 1.7710 1.7621 1.7533 1.7445 1.7357 1.7269
Columns 71 through 77
1.7181 1.7093 1.7004 1.6916 1.6827 1.6738 1.6649
Columns 78 through 84
1.6560 1.6469 1.6379 1.6287 1.6195 1.6103 1.6009
Columns 85 through 91
1.5915 1.5820 1.5724 1.5627 1.5529 1.5429 1.5329
Columns 92 through 98
1.5227 1.5124 1.5020 1.4915 1.4807 1.4699 1.4589
Columns 99 through 100
1.4477 1.4363
error_central =
Columns 1 through 7
7.5717 7.5509 7.5303 7.5101 7.4901 7.4704 7.4509
Columns 8 through 14
7.4318 7.4129 7.3943 7.3760 7.3579 7.3401 7.3226
Columns 15 through 21
7.3054 7.2884 7.2717 7.2553 7.2391 7.2232 7.2075
Columns 22 through 28
7.1921 7.1769 7.1620 7.1473 7.1329 7.1187 7.1047
Columns 29 through 35
7.0909 7.0774 7.0641 7.0510 7.0381 7.0255 7.0130
Columns 36 through 42
7.0007 6.9886 6.9767 6.9650 6.9535 6.9421 6.9309
Columns 43 through 49
6.9199 6.9090 6.8982 6.8877 6.8772 6.8669 6.8567
Columns 50 through 56
6.8467 6.8367 6.8269 6.8172 6.8076 6.7980 6.7886
Columns 57 through 63
6.7793 6.7700 6.7608 6.7517 6.7426 6.7336 6.7246
Columns 64 through 70
6.7157 6.7068 6.6979 6.6891 6.6802 6.6714 6.6626
Columns 71 through 77
6.6538 6.6450 6.6362 6.6273 6.6185 6.6096 6.6007
Columns 78 through 84
6.5917 6.5827 6.5736 6.5645 6.5553 6.5460 6.5367
Columns 85 through 91
6.5273 6.5177 6.5081 6.4984 6.4886 6.4787 6.4686
Columns 92 through 98
6.4585 6.4482 6.4378 6.4272 6.4165 6.4056 6.3946
Columns 99 through 100
6.3834 6.3721
b_right =
0
0
1
0
0
0
right_differencing =
Columns 1 through 7
-6.8689 -6.9403 -6.9668 -6.9540 -6.9068 -6.8300 -6.7277
Columns 8 through 14
-6.6038 -6.4618 -6.3047 -6.1354 -5.9564 -5.7700 -5.5782
Columns 15 through 21
-5.3827 -5.1853 -4.9872 -4.7897 -4.5937 -4.4004 -4.2103
Columns 22 through 28
-4.0241 -3.8424 -3.6657 -3.4942 -3.3283 -3.1680 -3.0137
Columns 29 through 35
-2.8653 -2.7228 -2.5864 -2.4558 -2.3311 -2.2121 -2.0987
Columns 36 through 42
-1.9907 -1.8880 -1.7903 -1.6976 -1.6096 -1.5261 -1.4469
Columns 43 through 49
-1.3718 -1.3006 -1.2332 -1.1692 -1.1086 -1.0511 -0.9966
Columns 50 through 56
-0.9449 -0.8958 -0.8492 -0.8049 -0.7628 -0.7227 -0.6846
Columns 57 through 63
-0.6482 -0.6135 -0.5805 -0.5489 -0.5187 -0.4899 -0.4622
Columns 64 through 70
-0.4358 -0.4105 -0.3862 -0.3629 -0.3406 -0.3191 -0.2986
Columns 71 through 77
-0.2789 -0.2599 -0.2418 -0.2244 -0.2077 -0.1918 -0.1765
Columns 78 through 84
-0.1619 -0.1480 -0.1348 -0.1221 -0.1102 -0.0988 -0.0881
Columns 85 through 91
-0.0780 -0.0686 -0.0597 -0.0515 -0.0439 -0.0368 -0.0305
Columns 92 through 98
-0.0247 -0.0195 -0.0149 -0.0110 -0.0076 -0.0049 -0.0027
Columns 99 through 100
-0.0012 -0.0003
error_right =
Columns 1 through 7
1.9332 2.0046 2.0311 2.0182 1.9711 1.8943 1.7920
Columns 8 through 14
1.6681 1.5260 1.3690 1.1996 1.0207 0.8342 0.6424
Columns 15 through 21
0.4470 0.2495 0.0514 0.1461 0.3420 0.5354 0.7255
Columns 22 through 28
0.9116 1.0933 1.2701 1.4415 1.6075 1.7677 1.9221
Columns 29 through 35
2.0705 2.2129 2.3494 2.4799 2.6047 2.7237 2.8371
Columns 36 through 42
2.9451 3.0478 3.1454 3.2381 3.3261 3.4096 3.4888
Columns 43 through 49
3.5639 3.6351 3.7026 3.7665 3.8271 3.8846 3.9391
Columns 50 through 56
3.9908 4.0399 4.0866 4.1309 4.1730 4.2130 4.2512
Columns 57 through 63
4.2875 4.3222 4.3553 4.3869 4.4170 4.4459 4.4735
Columns 64 through 70
4.4999 4.5253 4.5496 4.5728 4.5952 4.6166 4.6372
Columns 71 through 77
4.6569 4.6758 4.6939 4.7113 4.7280 4.7440 4.7592
Columns 78 through 84
4.7738 4.7877 4.8010 4.8136 4.8256 4.8369 4.8476
Columns 85 through 91
4.8577 4.8672 4.8760 4.8843 4.8919 4.8989 4.9053
Columns 92 through 98
4.9111 4.9163 4.9208 4.9248 4.9281 4.9309 4.9330
Columns 99 through 100
4.9345 4.9354
a_left =
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-5.0000 -4.0000 -3.0000 -2.0000 -1.0000 0
12.5000 8.0000 4.5000 2.0000 0.5000 0
-20.8333 -10.6667 -4.5000 -1.3333 -0.1667 0
26.0417 10.6667 3.3750 0.6667 0.0417 0
-26.0417 -8.5333 -2.0250 -0.2667 -0.0083 0
b_left =
0
0
1
0
0
0
>>
>> Plot will be
OBSERVATION:
- It has been concluded that the errors in central differencing method is much less than the other two schemes.
- The left-sided skewed and right sided skewed schemes shows nearly same amount of errors.
Why Skewed scheme is useful?
The skewed scheme is useful because it is necessary to compute the grids at the corners also which central differecing scheme is unable to do. Symmetric nodes does not present on the corners so that CD can not be implemented in such conditions.