1. For a point moving on a circle at constant angular velocity, w, the r and y coordinates are given by dy 2 71 dt = -wy. dt = WIT, W = Solve the above equations using a) the forward Euler scheme and b) the centred-difference scheme with a forward Euler first step. Use the following parameters: T' = 72h, At = 0.5h, To = (t = 0) = 600km, yo =0. Integrate for 141 h and plot the z, y coordinates on a polar plot. Comment on your results. NOTES: From these equations in cartesian coordinates (x, y) you can easily deduce that they are simpler in polar coordinates (r, 0). Since r2 =2 + y' you can multiply the a equation by z and the y equation by y yielding dr2/dt = 0. It also follows that de/ dt = w. In other words, the analytical solution is a point tracing out a circle of radius 600 km. In a) you are asked to integrate the 2 equations using forward Euler In1 = In - Atwyn Un+1 = yn + Atwin, which is a first-order scheme. The numerical result spirals outwards with radius increasing with time. This is clearly not accurate. In part b) this is replaced by centred differences Ent1= In-1 - 2Atwyn Un+1 = yn-1+ 2Atwin, (where you need to use the Forward Euler scheme above for the first time step) which is 2nd-order. The results are clearly better with the radius staying nearly constant