FDM by centred difference gives a gradient of -0.5 instead of -2

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Eustace Tan 2014-5-2

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链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/127907-fdm-by-centred-difference-gives-a-gradient-of-0-5-instead-of-2

Okay, maybe I'm just stupid, but the title pretty much says it all. The code I made is as follows,

n=[10:10:90 100:100:900 1000:1000:9000 10000:10000:90000];
for i=1:length(n),
      h(i)=1/(n(i));
      E=sparse(1:n(i)+1,1:n(i)+1,2*ones(1,n(i)+1),n(i)+1,n(i)+1);
      F=sparse(2:n(i)+1,1:n(i),-1*ones(1,n(i)),n(i)+1,n(i)+1);
      A=F+E+F';
      A(1,1)=1;
      A(1,2)=0;
      A(n(i)+1,n(i))=-1;
      A(n(i)+1,n(i)+1)=1;
      x=0:h(i):1;
      B(1,1:1+n(i))=((h(i))^2)*10;
      B(1,1)=0;
      B(1,1+n(i))=0;
      u=A\B';
      y=-5*x.^2+10*x; %Analytical solution
      e(i)=norm(u-y');
  end
figure
loglog(h.^-2,e,'r-+')
grid on
title('Graph of partition width to the power of -2, h^-^2 versus error, e')
xlabel('partition width to the power of -2, h^-^2')
ylabel('error, e')