A pulse advects at constant positive speed obeying

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

A pulse advects at constant positive speed cc obeying ut+cux=0u_t + c u_x = 0. Instead of the upwind (backward-difference) scheme, someone builds a "downwind" scheme by differencing forwards in space: ujn+1−ujnΔt+c uj+1n−ujnΔx=0.\frac{u_j^{n+1}-u_j^n}{\Delta t} + c\,\frac{u_{j+1}^n-u_j^n}{\Delta x} = 0. Let C=cΔt/ΔxC = c\Delta t/\Delta x be the Courant number, and treat uu as a smooth function so that Taylor expansion about (xj,tn)(x_j,t^n) is valid. ((a)) By Taylor-expanding both sides, dividing by Δt\Delta t, and eliminating the leading time derivatives using the relation utt=c2uxxu_{tt}=c^2u_{xx} (obtained by differentiating ut=−cuxu_t=-cu_x), show that the scheme's modified equation is ut+cux=Dnum uxx+O(Δx2,Δt2),Dnum=−12cΔx(1+C).u_t + cu_x = D_{\rm num}\,u_{xx} + O(\Delta x^2,\Delta t^2), \qquad D_{\rm num} = -\tfrac12 c\Delta x(1+C). ((b)) Evaluate DnumD_{\rm num} for c=2.0 m s−1c = 2.0\,\mathrm{m\,s^{-1}}, Δx=0.1 m\Delta x = 0.1\,\mathrm{m} and C=0.5C = 0.5. ((c)) State what the sign of DnumD_{\rm num} implies for the behaviour of the scheme at every Courant number C>0C>0, however fine the grid.

Browse the problem library