part a and b

3. Consider the BVP for the advection equation Ut + cux = 0, x ER, (c > 0), with u(x,0) = f(x), x > 0, u(0,t) = g(t), t > 0. a) Using the three-point molecule (@j , tn), (2j-1, tn+1), (Xj, tn+1), derive the implicit difference equation approximation (1+s)U+1 sum+1 = Um, b) Show that the scheme is von Neumann stable. c) Develop an algorithm to compute Um at lattice points in the first quad- rant. Explain carefully your procedure. 3. Consider the BVP for the advection equation Ut + cux = 0, x ER, (c > 0), with u(x,0) = f(x), x > 0, u(0,t) = g(t), t > 0. a) Using the three-point molecule (@j , tn), (2j-1, tn+1), (Xj, tn+1), derive the implicit difference equation approximation (1+s)U+1 sum+1 = Um, b) Show that the scheme is von Neumann stable. c) Develop an algorithm to compute Um at lattice points in the first quad- rant. Explain carefully your procedure