Growth estimates of the Crank-Nicolson scheme

    Activity: Talk or presentationOral presentation


    Given the linear, time-invariant, differential equation: $\dot{x}(t) = A x(t)$ together with its Crank-Nicolson Approximation: $x_d(n+1) = (I + \Delta A/2)(I - \Delta A/2)^{-1} x_d(n)$, where $\Delta$ is the discretization step. It is well-known that if all the eigenvalues of $A$ have negative real part, then all eigenvalues of $A_d:=(I + \Delta A/2)(I - \Delta A/2)^{-1}$ lie within the unit circle. Hence for matrices, stability of the differential equation carries over to its approximation. In this talk we concentrate on the relation between the overshoots, i.e. $M_c := \sup_{t \geq 0} \|\exp(At} \|$ versus $M_d = \sup_{n \in N} \| A_d^n \|$. We show that for a non-Euclidean norm $M_d$ will depend on the $M_c$ and on the dimension of the vector $x$. For the Euclidean norm the situation is less clear. Among others, we show that it is related to properties of $exp(A^{-1}t)$.
    Period25 Jan 2012
    Held atCentre for Analysis, Scientific computing and Applications (CASA), Netherlands
    Degree of RecognitionNational