3. ODE Solution Methods [10 marks] Consider the differential equation, initial condition, and analytical solution below

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3 Ode Solution Methods 10 Marks Consider The Differential Equation Initial Condition And Analytical Solution Below 1 (142.77 KiB) Viewed 36 times

3. ODE Solution Methods [10 marks] Consider the differential equation, initial condition, and analytical solution below X' = -X + t, to, Xo = [0,0], = X = e-t+t-1 The calculation of one Euler step is given in the table below for At = 0.1. It is rather trivial because the derivative is zero for this step. i Xi f(ti, x;) ti+1 Xi+1 0 0 0 0 0.1 0 1 We seek, x(0.2), an estimate of X at t = 0.2. The analytical solution is 0.0187. (a) Calculate the estimate by taking a second Euler step with At = 0.1 [2 marks] (b) Calculate the estimate using one Euler step with At = 0.2 [1 mark] (c) Calculate the estimate using Richardson Extrapolation [2 marks] (d) Calculate the absolute errors for the three estimates above and state how they are related. [3 marks] (e) Speculate on the size of the errors for At = 2.1 [2 marks]