We have a hyperbolic pde (in fact the 1d wave equation) with indep vars X, T We use the central difference approximations for the second derivatives wrt X, T to get [phi(Xn, Tj+1) -2phi(Xn, Tj) + phi(Xn, Tj+1)]/(dT^2) = [c^2][phi(Xn-1, Tj) -2phi(Xn, Tj) + phi(Xn=1, Tj)]/(dX^2) where dX and dT are the changes in the independent variables between steps, c is the speed of the wave. We are supposed to show that the scheme is stable only if cdT/dX = lambda<=1 We have to apply Von Neumann stability analysis. This usually means assuming a solution of the form phi = A(T)*exp(ikX) Xn = ndX therefore phi(Xn, Tj) = Aj*exp(ikndX) The stability condition is that sigma = Aj+1/Aj and we want the modulus of sigma is less than or equal to one. The first problem I have is that the above PDE finite difference equation has phi's with three different times in it; j-1,j,j+1. Therefore I tried using two differenct sigma factors sigma1 = Aj/Aj-1 and sigma2 = Aj+1/Aj I do not think that you can assume that these are the same. I think that for stability you need the modulus of both sigmas is less than or equal to one. the original finite difference equation gives us sigma1 = 1/[2(lambda^2)cos(kdX) +2(1-(lambda^2)) - sigma2) I set the modulus of this is less than or equal to one (call this stage A), giving two more equations: -2(lambda^2)cos(kdX) - 2 + 2(lambda^2) + sigma2 <=1 this must be true for the largest values of the LHS (i.e when sigma2 = 1) , giving us Lambda^2 <= 1/(1-cos(kdX)) The RHS is largest when cos = 0, and also note that dX, dT, c are all positive by def'n, giving us the required condition. The only problem is we also get another equation from stage A: 2(lambda^2)cos(KdX) + 2 - 2(lambda^2) - sigma2 >=1 This must be true for the smallest values of the LHS i.e when sigma2 = 1 This implies that (lambda^2)(cos(kdX) - 1)>=0 We know that lambda^2>=0 which means that the other bracketed term must be greater than or equal to one. This is only satisfied for cos = 0 This extra condition was not mentioned in the question and doesn't look right. I'm not sure where I've gone wrong.