How to include piecewise defined functions in bvp4c solver in Matlab - matlab

Question

I am having trouble using the bvp4c with piecewise defined functions.
I tested the code and it works fine when the piecewise defined functions are constant.
The problem is that I get wrong results in the graph (that I know for sure) in the area where the piecewise defined functions are not constant.
Any ideas or suggestions on how to handle this problem?
Thanks
function bvp4
xlow=0;
xhigh=0.30;
solinit=bvpinit(linspace(xlow,xhigh,1000),[0 0]);
sol = bvp4c(#bvp4ode,#bvp4bc,solinit);
xint=[xlow:0.0001:xhigh];
Sxint=deval(sol,xint);
Sxint1=abs(sqrt(Sxint));
xint=[xlow:0.0001:xhigh];
plot(xint,Sxint1(1,:),'r')
function dydx = bvp4ode(x,y)
So=0.00125;
s=1.5;
dydx = [y(2);
((G(x)+125*f(x)*y(1)*(1+1/s^2)^0.5-1000*9.81*So*H(x))/(1000*0.5*l(x)*(f(x)/8)^0.5)-y(2)*2*(-2/3*x+0.071+2/3*0.08)*(-2/3)*b(x))/H(x)/H(x)];
function res = bvp4bc(ya,yb)
res = [ya(1); yb(1)];
function fval = f(x)
if (x >= 0) && (x <= 0.08)
fval = 0.0187;
elseif (x > 0.08) && (x <= 0.17)
fval = 0.0298;
elseif (x > 0.17) && (x <= 0.3)
fval= 0.0408;
end
function Gval = G(xint)
if (xint >= 0) && (xint <= 0.08)
Gval = 0.1306;
elseif (xint > 0.08) && (xint <= 0.17)
Gval = 0.1306;
elseif (xint > 0.17) && (xint <= 0.3)
Gval = -0.0337;
end
function Hval = H(xint)
if (xint >= 0) && (xint < 0.08)
Hval = 0.071;
elseif (xint >= 0.08) && (xint <= 0.17)
Hval = -2/3*xint+(0.071+2/3*0.08);
elseif (xint >0.17) && (xint <= 0.3)
Hval = 0.011;
end
function bval = b(xint)
if (xint >= 0) && (xint < 0.08)
bval = 0;
elseif (xint >= 0.08) && (xint <= 0.17)
bval = 1;
elseif (xint > 0.17) && (xint <= 0.3)
bval= 0;
end
function lval = l(xint)
if (xint >= 0) && (xint <= 0.08)
lval = 0.067;
elseif (xint > 0.08) && (xint <= 0.17)
lval = 0.134;
elseif (xint > 0.17) && (xint <= 0.3)
lval= 1.165;
end

Answer 1

Thou shalt not surprise your fragile solver with sudden jumps.
Any order p solver expects an ODE function that is at least p times continuously differentiable to sensibly adapt the mesh of gridpoints resp. the local step size. Any deviation leads to excessive and possibly oscillating adaptation near the singular points, resulting in long computation times or underflow in the step size.
I see two possibilities to work around this problem, use events (if supported for BVP) to switch models/ODE functions or use the provided multipoint mechanism to split the integration interval into sections where your parameter functions are constant. Then you can also use simple arrays for the parameters instead of functions with many branchings.