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A bounded approximate Jacobian can destroy Rosenbrock stiff damping

Statement

For the scalar family $y^{'} = - λ y$ with $λ > 0$ and $y_{0} = 1$ , take $h = 4 / λ$ , $\tilde{J} = 0$ , and $\tilde{T} = 0$ in the exact-arithmetic stage equations of the two-stage method. Then $W = 1$ , its amplification factor is $5$ , and this approximate-J method amplifies the mode even though the exact-J method at $z = - 4$ has modulus less than one. The zero approximation is bounded. This calculation demonstrates loss of stiff damping and makes no order or uniform-convergence claim.

Facts & Assumptions

Given: The scalar equation, $λ > 0$ , $h = 4 / λ$ , and the stated approximate derivatives.

[F1]

Two-stage second-order Rosenbrock W-method with shared factorization: the stages satisfy $W k_{1} = f + α h \tilde{T}$ , $F_{1} = f (y + h k_{1} / 2)$ for this autonomous equation, $W q_{2} = F_{1} - k_{1}$ , $k_{2} = q_{2} + k_{1}$ , and $y_{+} = y + h k_{2}$ .

[F2]

The selected exact-J two-stage Rosenbrock method is scalar L-stable: for the exact Jacobian, $R (z) = [1 + (1 - 2 α) z] / (1 - α z)^{2}$ and it is L-stable.

Proof

Given: The data in the Statement.

1.1givenF1algebra

Here $W = 1$ and $k_{1} = f (y) = - λ y$ . Since $h k_{1} / 2 = - 2 y$ , the midpoint state is $- y$ and $F_{1} = f (- y) = λ y$ .

2.1step 1.1F1algebra

The second right-hand side is $F_{1} - k_{1} = 2 λ y$ , so $q_{2} = 2 λ y$ and $k_{2} = q_{2} + k_{1} = λ y$ . Therefore $y_{+} = y + h λ y = 5 y$ , giving amplification factor $5$ .

3.1givenF2step 2.1algebra

For the exact-J branch, [F2] gives $R (- 4) = [1 - 4 (1 - 2 α)] / (1 + 4 α)^{2} = (5 - 4 \sqrt{2}) / (5 - 2 \sqrt{2})^{2}$ , because $α = 1 - 1 / \sqrt{2}$ . Since $25 < 32 < 36$ , we have $5 < 4 \sqrt{2} < 6$ , so $∣ 5 - 4 \sqrt{2} ∣ < 1$ ; also $2 \sqrt{2} < 4$ , so $5 - 2 \sqrt{2} > 1$ . Thus $∣ R (- 4) ∣ < 1$ . This differs from step 2.1 and proves that boundedness of $\tilde{J}$ alone does not preserve stiff damping.

4.1step 1.1step 2.1algebra∎

The approximate-J calculation in step 2.1 and the exact-J factor comparison establish the stated damping failure. No order or convergence conclusion follows from this stiff-scaled calculation.

Finite check

With $λ = 4$ and $h = 1$ , the approximate-J output is $5 y$ , while the exact-J factor is $(5 - 4 \sqrt{2}) / (5 - 2 \sqrt{2})^{2} \approx - 0.1393$ .

Depends on

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Sources

WIAS Berlin, Numerical Methods for Stiff Ordinary Differential Equations, §2.4 Rosenbrock Methods (standard reference, not scraped)
Daniel Kressner, Advanced Numerical Analysis, §3.5 Rosenbrock methods (standard reference, not scraped)
Joachim Rang and Lutz Angermann, New Rosenbrock Methods of Order 3 for PDAEs of Index 2 (standard reference, not scraped)