I believe you can solve this particular equation analytically. For if F(x,t)=f(x,t)/x x\partial_t F(x,t)+\partial_x F(x,t)=0 is satisfied by F(x,t)=phi(t-.5x^2), where phi is set by e.g. the boundary condition at x_0 or the initial condition at t_0 Then f(x,t)=phi(t-.5x^2)/x + HTH. George Nurser.