1. Stability function h = 0.7 h = 0.3 Model Problem Exact solution
Use Euler method with h = 0.7
and h = 0.3
h = 0.7
clear
a = 0; b = 4; n = 20; y0 = 1;
f = inline('-(3).*y','x','y');
exact = inline('exp(-(3).*x)'); %%
h = 0.7;
x = a:h:b;
y(1) = y0;
ex(1) = y0;
%% Euler steps
for i=1:length(x)-1
y(i+1) = y(i) + h * f(x(i),y(i));
ex(i+1) = exact(x(i+1));
end
plot(x,y,'-ob',x,ex,'-*r');
grid on
legend('euler','exact');
xlabel('x');
ylabel('y');
h = 0.3

2. Stability function Model Problem Explicit Euler Method Exact solution
To study the stability of a method, you need to apply your method to this model problem
Exact solution
this stability region is the disc with centre at −1 and radius 1.

3. Stability function for RK
Sx1 vector
Sx1 vector
ones vector
Size of what
Size of what

4. Stability function for RK

5. Explicit Runge–Kutta method
Stability function for RK
Explicit Runge–Kutta method
Geometric series
the order condition

6. Stability function for RK
clear
% RK41
b=[1/6;1/3;1/3; 1/6];
c=[0;1/2; 1/2;1];
A=[ ;1/ ;0 1/2 0 0; ]; %
% R(z) = 1 + z b^T ( I - z A )^(-1) 1
format rat
r2 = b'*A^0*c, r3 = b'*A^1*c, r4 = b'*A^2*c
r5 = b'*A^3*c, r6 = b'*A^4*c, r7 = b'*A^5*c
r8 = b'*A^6*c
r2 = 1/2
r3 = 1/6
r4 = 1/24
r5 = 0
r6 = 0
r7 = 0
a=3;
b=3;
x=-a:0.1:a;
y = -b:0.1:b;
[x,y]=meshgrid(x,y);
z = x + 1i*y;
R = 1 + z + r2*z.^2 + r3*z.^3 + r4*z.^4 + r5*z.^5 + r6*z.^6;
contour(x,y,abs(R),[0.001:0.005:1]);colorbar
hold on;
line([-a,a],[0,0])
line([0,0],[-b,b])
hold off
grid on

7. Stability function for RK
clear
% RK41
b=[1/6;1/3;1/3; 1/6];
c=[0;1/2; 1/2;1];
A=[ ;1/ ;0 1/2 0 0; ]; %
% % RK42
% b=[1/6;0;2/3; 1/6];
% c=[0;1/4; 1/2;1];
% A=[ ;1/ ;0 1/2 0 0; ];
% % %
% % RK5
% b=[7/90;0;32/90; 12/90;32/90;7/90];
% c=[0;1/4; 1/4;1/2;3/4;1];
% A=[ ;1/ ;1/8 1/ ;0 0 1/ ;3/16 -3/8 3/8 9/16 0 0;-3/7 8/7 6/7 -12/7 8/7 0];
% R(z) = 1 + z b^T ( I - z A )^(-1) 1
format rat
r2 = b'*A^0*c, r3 = b'*A^1*c, r4 = b'*A^2*c
r5 = b'*A^3*c, r6 = b'*A^4*c, r7 = b'*A^5*c
r8 = b'*A^6*c
% b'*A^7*c
% b'*A^8*c
% b'*A^10*c
a=3;
b=3;
x=-a:0.1:a;
y = -b:0.1:b;
[x,y]=meshgrid(x,y);
z = x + 1i*y;
R = 1 + z + r2*z.^2 + r3*z.^3 + r4*z.^4 + r5*z.^5 + r6*z.^6;
contour(x,y,abs(R),[1,1]);colorbar
hold on;
line([-a,a],[0,0])
line([0,0],[-b,b])
hold off
grid on

8. Stability function for IRK

9. Stability function for IRK
one_v =ones(s,1)
syms z
NN = det(eye(s)+z*(one_v*b'-A))
DD = det(eye(s)-z*A)
solve(DD)
D = inline(DD)
N = inline(NN)
E = D(1i*y)*D(-1i*y) - N(1i*y)*N(-1i*y)
a=40;
b=20;
x=-a:0.1:a;
y = -b:0.1:b;
[x,y]=meshgrid(x,y);
z = x + 1i*y;
R = subs(NN/DD);
contour(x,y,abs(R),[0.1,0.4,0.7,1]);
hold on;
line([-a,a],[0,0])
line([0,0],[-b,b])
hold off
grid on

10. Stability Padé approximant degree l degree m
Approximte f(z) by rational function

11. Stability

12. Stability
Linear stability analysis is based on the linear test problem
1) non-autonomous
generalizing this analysis in two possible ways
1) non-autonomous
2) Non-linear stability

13. Stability

14. Stability
Example:
is it A-stable
is it AN-stable

15. Stability
Definition:
L-stable
the method is A-stable and that, in addition,