Task 1 Consider the following schemes for the heat equation un+1 _ un-1 Leapfrog scheme: m = d'm+1 - 20m + um Um-1 + fm 2k h2 1.1 Show that the Leapfrog scheme is consistent with the heat equation. 1.2 Use the von Neumann stability analysis to show that the amplification polynomial for the Leapfrog scheme is given by 92 + 8gdusin2 50 - 1 = 0, where A = k Use the Schur and von Neumann theory to comment on the stability of the scheme. Task 2 Show that a scheme for the heat equation ut - durr = 0, for (x, t) E R x (0, T), of the form 1 -a 2 (Um+1 +Um-1), with a constant as h and & tend to zero, is consistent with the heat equation only if a = 1 - 2du