implicit-explicit method for finite difference method under reaction diffusion equataions

Hello, I am trying to solve the following reaction diffusion numerically under implicit-explicit method.

I try to compare my numerical solution to artifical true solution by define

and

The implicit -explicit scheme is straight forward and I will skip the derivation. However, the error between numerical and analytic solution does not follow the first order as dx goes to 0. I am wondering where I made a small but stupid mistake. Codes are attached. Thanks for any comment.

L = 50; % spatial size

Time = 50; % final time

delta = 0.05;

index=0:1:5;

h=power(3,-index);

forward_true=@(time,space) (cos(pi.*space/50).^2-1/2).*exp(0.01*time);

f_true=@(dk) (0.01+4*(pi/50).^2+delta)*dk;

error=zeros(1,length(index));

for test=1:length(index)

dx=h(test);

dt=dx;

x=0:dx:L;

t=0:dt:Time;

A=1;

N=length(t);

M=length(x);

mu = dt/dx/dx;

[X,T] = meshgrid(x,t); % This will create two matrices, T and X, where each element of t and x is paired with the other.

data=forward_true(T,X);

K0=cos(pi.*x/50).^2-1/2;

solu=zeros(N,M);

solu(1,:)=K0;

% implicit method

main_diag=1+2*mu+delta*dt;

upper=-mu;

lower=-mu;

Mdiag=main_diag*ones(M,1);

Muppder=upper*ones(M-1,1);

Mlower=lower*ones(M-1,1);

matrix=diag(Mdiag)+diag(Muppder,1)+diag(Mlower,-1);

matrix(1,2)=-2*mu;

matrix(end,end-1)=-2*mu;

for tstep=1:N-1

Kc = solu(tstep,:)';

Knext=zeros(1,M);

fvalue = f_true(Kc);

RHS=Kc+dt*fvalue;

Knext=matrix\RHS;

solu(tstep+1, :) = Knext;

end

error(test)=norm(data-solu,inf);

disp(test)

end

[Yinxi Pan]