Computation of global error bound for Euler's formula

Question

I am trying to calculate the global error bound for Euler's method, but I am having trouble. I am given the formula $|y(t_{i}) - u_{i}| \leq \frac{1}{L}(\frac{hM}{2} + \frac{\delta}{h})(e^{L(t_{i}-a)}-1) +|\delta_{0}|e^{L(t_{i}-a)}$ where $u_{i}$ is the Euler approxmation. I am also given $M, L, a, \delta, \delta_{0}$, h. If I am not mistaken this will give me the error for each step, but how do I find the upper bound for the total error?

Answer 1

For the error $e_k=|y(t_k)-u_k|$ you get the inequality $$ e_{k+1}\le e_k+h|f(y(t_k))-f(u_k)|+l_k\le (1+Lh)e_k+l_k $$ where $l_k$ contains all the local errors, the method step error $$|y(t_{k+1})-y(t_k)-hf(y(t_k))|\approx \frac{h^2}2|y''(t_k)|, $$ and the floating point evaluation errors. The recursion inequality can be solved to $$ e_n \le (1+Lh)^ne_0+\frac{(1+Lh)^n-1}{Lh}\max_{0\le k<n}l_k $$ Now use $(1+Lh)^n\le e^{Lnh}$ and the bounds for the initial error and the local step errors to get the claimed result. Use the Taylor formula or mean value formula to get proper bounds on the method step truncation error.