Math 6610 • Analysis of Numerical Methods I

Compare the stability of the Euler Predictor-Trapezoidal Corrector and Backwards Euler timestepping methods for solution of the heat equation \[ u_t=u_{xx}\qquad u(x,0)=u_0(x) \] using the centered second difference operator \[ u_j \to \frac {u_{j+1}-2u_j+u_{j-1}}{h^2}\qquad j=1,\ldots ,J \] where \(u_j\approx u(jh)\) on the interval \([0,2\pi ]\) with periodic boundary conditions, and mesh-size \(h=2\pi /J\). Discuss the accuracy of these methods?