Environmental and Biological Modelling in Maths

MAS305 Environmental and Biological Modelling Semester 1 2015 Assignment 4 (Due: June 1) 1. Consider the linearised model for the height of the phreatic surface in a porous 10m wide levee separating two ponds (both initially at depth 5m) ∂h1 ∂t = Kh0 ∂ 2h1 ∂x2 for 0 < x < 10 where h(x, t) = h0 + h1(x, t) (h0 is the constant reference height). Suppose that at t = 0 the heights on the ponds on the two sides of the levee are changed to h(0, t) = 4m and h(10, t) = 6m and we are interested in the effect this has on the height of the phreatic surface. (a) What would be an appropriate value for h0 in this problem? (b) Formulate the IBVP for h1(x, t). (c) Find the steady state solution for h1 a long time after the pond levels have changed. (d) Assume K = 0.4 and use separation of variables (after subtracting the steady state solution) to find the full solution for h1(x, t). (e) Using MATLAB or similar plot a few profiles of h1(x, t) to show the approach to steady state. 2. Consider the numerical solution of the advection equation ∂T ∂t + U ∂T ∂x = 0, U > 0 with the discretisation xj = j∆x, tn = n∆t and T n j = T(xj , tn). (a) Using a forward difference in time and a central difference in space to derive the finite difference equation T n+1 j = T n j − c 2 (T n j+1 − T n j−1 ) where c = U∆t/∆x. (b) By considering the propagation of the single Fourier component of the error ε n j = G n exp(ijkπ∆x) show that G is given by G = 1 − ic sin(kπ∆x). (Hint: substitute ε n j in the finite difference equation and rearrange.) (c) Hence deduce that this method is unstable for all c > 0. (d) Redo the analysis above but use a backward difference in space to give the finite difference equation T n+1 j = T n j − c(T n j − T n j−1 ). (e) Show that in this case |G| 2 is given by |G| 2 = 1 − 2c(1 − c)(1 − cos(kπ∆x)) and hence deduce that this method is stable for c ∈ [0, 1]. 3. Consider the function φ(x, y) = x − x 2 + y 2 (a) Show that φ(x, y) is a harmonic function (i.e. it satisfied Laplace’s equation). (b) Find ψ(x, y) so that F(z) = φ(x, y) + iψ(x, y) is an analytic function. (c) Express F(z) explicitly as a function of z (ie no x’s or y’s).