A Numerical Framework for Integrating Deferred Correction Methods to Solve High Order Collocation Formulations of ODEs

Abstract

Recent analysis and numerical experiments show that the deferred correction methods are competitive numerical schemes for time dependent differential equations. These methods differ in the mathematical formulations, choices of collocation points, and numerical integration or differentiation strategies. Existing analyses of these methods usually follow traditional ODE theory and study each algorithm’s convergence and stability properties as the step size $\varDelta t$ varies. In this paper, we study the deferred correction methods from a different perspective by separating two different concepts in the algorithm: (1) the properties of the converged solution to the collocation formulation, and (2) the convergence procedure utilizing the deferred correction schemes to iteratively and efficiently reduce the error in the provisional solution. This new viewpoint allows the construction of a numerical framework to integrate existing techniques, by (1) selecting an appropriate collocation discretization based on the physical properties of the solution to balance the time step size and accuracy of the initial approximate solution; and by (2) applying different deferred correction strategies for reducing different components in the error of the provisional solution. This paper discusses properties of different components in the numerical framework, and presents preliminary results on the effective integration of these components for ODE initial value problems. Our results provide useful guidelines for implementing “optimal” time integration schemes for general time dependent differential equations.

Appendix

Proof of Theorem 3

Assuming p points $\{ 1/p, 2/p, \dots , (p-1)/p, 1 \}$ are used in the uniform collocation formulation, then $\tilde{S}$ is a lower triangular matrix and all non-zero entries (including diagonal entries) are 1 / p. Simple calculation shows that $\tilde{S}^{-1}$ has zero entries everywhere except along the diagonal and subdiagonal, with nonzero entries p on the diagonal and $-p$ on the subdiagonal,

$$\begin{aligned} \tilde{S}^{-1} = \left[ \begin{array}{llllll} p &{}\quad 0 &{} \quad 0 &{} \quad \cdots &{}\quad 0 &{}\quad 0 \\ -p &{}\quad p &{} \quad 0 &{}\quad \cdots &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad -p &{} \quad p &{} \quad \cdots &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -p &{}\quad \cdots &{} \quad 0 &{}\quad 0 \\ \cdots &{} \quad \cdots &{}\quad \cdots &{} \quad \cdots &{}\quad \cdots &{}\quad \cdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad -p &{}\quad p \end{array} \right] . \end{aligned}$$

Consider the vector $\mathbf{V}_j = [ (p-1)^j, (p-2)^j, \dots , 2^j, 1^j, 0]^T$ ($j=1,\dots ,p-1$) and $\mathbf{V}_0=[1,1, \ldots , 1,1]^T$. As S integrates polynomials of degree $\le p-1$ exactly, one can show

$$\begin{aligned} \left( \tilde{S}^{-1} S - I\right) \mathbf{V}_j = \frac{1}{j+1} \sum _{l=0}^{j-1} \left( {\begin{array}{c}l\\ j+1\end{array}}\right) \mathbf{V}_l \quad and \quad \left( \tilde{S}^{-1} S - I\right) \mathbf{V}_0 = \mathbf{0}. \end{aligned}$$

Define $\mathbf{W}_0=\mathbf{V}_0$. The basis for the Jordan canonical form can be constructed recursively by solving $(\tilde{S}^{-1} S - I) \mathbf{W}_j = \mathbf{W}_{j-1}$, where $\mathbf{W}_j$ consists of a linear combination of $\mathbf{V}_k$, $k=0, \ldots , j$. $\square$

To prove Theorem 4, we start from the following Lemma:

Lemma 1

For the trapezoidal rule preconditioned uniform collocation formulation (InDC-yp-T), the matrix $S-\tilde{S}$ maps the vector $[ (\frac{j}{p})^k ]_{j=0}^p := [ (\frac{0}{p})^k, (\frac{1}{p})^k, (\frac{2}{p})^k, \dots , (\frac{p-1}{p})^k, 1 ]^T$ ($k \le p$) to a linear combination of vectors $[ (\frac{j}{p})^m ]_{j=0}^p$, $m=0, \dots , k-1$.

Proof

Assume $p+1$ points $\{ 0/p, 1/p, 2/p, \dots , (p-1)/p, 1 \}$ are used in the uniform collocation formulation. As the integration matrix S integrates polynomials of degree p or less exactly, we have

$$\begin{aligned} S \left[ \left( \frac{j}{p}\right) ^k \right] _{j=0}^p = \left[ \int _0^{\frac{j}{p}} x^k dx \right] _{j=0}^p = \frac{1}{k+1} \left[ \left( \frac{j}{p}\right) ^{k+1} \right] _{j=0}^p. \end{aligned}$$

Now consider the $j^{th}$ entry of the vector $\tilde{S} [ (\frac{j}{p})^k ]_{j=0}^p$ given by

$$\begin{aligned} \tilde{S} \left[ \left( \frac{j}{p}\right) ^k \right] _j= & {} \frac{1}{p} \left( \frac{1}{2} \left( \frac{0}{p}\right) ^k + \sum _{n=1}^{j-1} \left( \frac{n}{p}\right) ^k + \frac{1}{2} \left( \frac{j}{p}\right) ^k \right) =\frac{1}{p^{k+1}}\left( \sum _{n=1}^j n^k - \frac{1}{2} j^k\right) \\= & {} \frac{1}{p^{k+1}}\left( \frac{j^{k+1}}{k+1} + \frac{1}{2} j^k + \text{ lower } \text{ order }\,(<k)\hbox { terms }-\frac{1}{2} j^k\right) . \end{aligned}$$

Therefore, after cancelling the $j^{k+1}$ and $j^k$ terms, we have

$$\begin{aligned} (S-\tilde{S})\left[ \left( \frac{j}{p}\right) ^k \right] _{j=0}^p = \sum _{m=0}^{k-1} c_m \left[ \left( \frac{j}{p}\right) ^m \right] _{j=0}^p. \end{aligned}$$

$\square$

Proof of Theorem 4

We will Apply Lemma 1 and the Taylor expansion of the initial provisional solution in the trapezoidal rule preconditioned deferred correction iterations for the uniform collocation formulation (InDC-yp-T). From Eq. (17), we see that the correction matrix has the expansion

$$\begin{aligned} C^t_{ns}=(\lambda \varDelta {t})\left( \mathbf S - \tilde{\mathbf{S }}\right) + (\lambda \varDelta {t})^2 \tilde{\mathbf{S }}\left( \mathbf S - \tilde{\mathbf{S }}\right) + (\lambda \varDelta {t})^3 \tilde{\mathbf{S }}^2\left( \mathbf S - \tilde{\mathbf{S }}\right) + \cdots , \end{aligned}$$

and the initial provisional solution b has the expansion of the form (neglecting all $(\varDelta {t})^{p+1}$ and higher order terms)

$$\begin{aligned} b \approx \sum _{m=0}^{p} (\lambda \varDelta {t})^m c_m \left[ \left( \frac{j}{p}\right) ^m \right] . \end{aligned}$$

By induction and Lemma 1, it is straightforward to show that

$$\begin{aligned} \left( C^t_{ns}\right) ^k b \approx (\lambda \varDelta {t})^{2k} \sum _{m=0}^{p} c_{m,k} \left[ \left( \frac{j}{p}\right) ^m \right] , \end{aligned}$$

neglecting $(\varDelta {t})^{p+1}$ and higher order terms. Therefore, after each trapezoidal rule preconditioned SDC iteration for the uniform collocation formation, the order will increase by $(\varDelta {t})^2$, until it reaches $(\varDelta {t})^{p+1}$. $\square$