Say we have a function , that satisfies the ordinary differential equation , where takes discrete values, , with a constant step size . We also are given the initial condition, for example, .
If we were to use the Backward Euler Method:
to approximate a solution for a function, say, , I am told that it is possible to use this information to show that the Backward Euler Scheme is first order, by expanding the global error as a Taylor series in powers of . (The global error at fixed is the difference between the approximate solution and the exact solution, )
Can anyone figure out how this might be possible? I cannot find any resources online that dmeonstrate this idea.