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Noncommuting two-by-two matrices distinguish Lie and Strang local errors

Example

On $R^{2}$ , let $a (x) = A x$ and $b (x) = B x$ , where $A = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), B = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}) .$ The fields are smooth and their exact flows exist globally. With the bracket convention in Lie and Strang compositions of exact split flows, the Lie split has a nonzero $h^{2}$ local defect and the symmetric Strang split has a nonzero $h^{3}$ local defect at $x = e_{1} = (1, 0)^{T}$ .

Facts & Assumptions

Given: The two matrices and the corresponding linear vector fields on $R^{2}$ .

[F1]

Lie and Strang compositions of exact split flows: for linear fields $a (x) = A x$ , $b (x) = B x$ , the bracket is $[a, b] (x) = (B A - A B) x$ .

[F2]

Lie and Strang splitting errors have commutator expansions: the Lie defect is $\frac{1}{2} h^{2} [a, b] (x) + O (h^{3})$ and the Strang defect has leading coefficient $- \frac{1}{24} [a, [a, b]] (x) - \frac{1}{12} [b, [a, b]] (x)$ at order $h^{3}$ .

[F3]

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes: matrix products are computed by row-by-column multiplication.

[F4]

The exponential series of a bounded operator: every finite matrix $M$ has an everywhere-defined exponential $e^{t M}$ with $e^{0} = I$ and derivative $M e^{t M}$ , so its linear ODE flow exists for all real $t$ .

Proof

1.1F1F3F4algebra

By [F4], the linear fields have global exact flows. Direct multiplication gives $B A = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}), A B = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}), C : = B A - A B = (\begin{array}{cc} - 1 & 0 \\ 0 & 1 \end{array}) .$ By [F1], $[a, b] (x) = C x$ .

2.1F1F3step 1.1algebra

Multiplying once more gives $C A - A C = (\begin{array}{cc} 0 & - 2 \\ 0 & 0 \end{array}), C B - B C = (\begin{array}{cc} 0 & 0 \\ 2 & 0 \end{array}) .$ These are the matrices of $[a, [a, b]]$ and $[b, [a, b]]$ under the same convention.

2.2F2step 1.1algebra

At $e_{1}$ , $C e_{1} = - e_{1} \neq 0$ , so [F2] gives Lie local defect $- \frac{1}{2} h^{2} e_{1} + O (h^{3})$ , whose leading coefficient is nonzero. Thus its local error has exact order two.

3.1F2step 2.1algebra∎

The Strang leading matrix from step 2.1 is $- \frac{1}{24} (C A - A C) - \frac{1}{12} (C B - B C) = (\begin{array}{cc} 0 & 1 / 12 \\ - 1 / 6 & 0 \end{array}) .$ It sends $e_{1}$ to $(0, - 1 / 6)^{T} \neq 0$ , so [F2] gives a nonzero cubic leading local defect and exact local order three.

Depends on

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Sources

Ernst Hairer, Geometric Numerical Integration, Lecture 2: Symplectic Integrators (standard reference, not scraped)