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Rosenbrock–Euler has an explicit one-step defect and solve-residual allowance

Statement

Let $D \subseteq R \times R^{d}$ be open and $f : D \to R^{d}$ be $C^{2}$ (componentwise, as in $C^{k}$ Euclidean maps and diffeomorphisms). Let $y (s)$ solve $y^{'} = f (s, y)$ on $[t, t + h]$ , $h > 0$ , and set $J = D_{y} f (t, y (t))$ . Suppose the trajectory lies in a region where $∥ f ∥_{2} \leq F$ , $∥ D_{y} f ∥_{2} \leq L$ , and $∥ f_{t} + D_{y} f f ∥_{2} \leq M_{1}$ . Fix $γ > 0$ and a real matrix $\tilde{J}$ . Suppose $A = I - γ h J$ and $\tilde{A} = I - γ h \tilde{J}$ are invertible with $∥ A^{- 1} ∥_{2} \leq κ_{0}$ and $∥ {\tilde{A}}^{- 1} ∥_{2} \leq κ_{1}$ . Write $δ = ∥ \tilde{J} - J ∥_{2}$ .

For the exact-arithmetic Rosenbrock stage map $Φ_{γ, h}^{\tilde{J}}$ , and a computed stage $\hat{k}$ whose update residual obeys $∥ \tilde{A} \hat{k} - f (t, y (t)) ∥_{2} \leq ρ$ , the update ${\hat{y}}_{+} = y (t) + h \hat{k}$ satisfies $∥ {\hat{y}}_{+} - y (t + h) ∥_{2} \leq C_{E} h^{2} + κ_{1} h ρ, C_{E} : = γ κ_{0} L F + \frac{M_{1}}{2} + γ κ_{0} κ_{1} δ F .$ The last term is the separate Jacobian-approximation allowance. The residual is an equation residual in the units of Implicit residuals and their update and derivative units, not a forward-error estimate by itself.

Facts & Assumptions

Given: The stated exact solution, derivative bounds, matrices, inverse bounds, and residual bound.

[F1]

The Rosenbrock–Euler step map with an exact or approximate Jacobian: $Φ_{γ, h}^{J} (t, y) = y + h (I - γ h J)^{- 1} f (t, y)$ .

[F2]

The one-stage update has an explicit Jacobian-approximation defect: $∥ Φ_{γ, h}^{\tilde{J}} (t, y) - Φ_{γ, h}^{J} (t, y) ∥_{2} \leq γ h^{2} κ_{0} κ_{1} δ ∥ f (t, y) ∥_{2}$ .

[F3]

Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: the induced Euclidean norm satisfies $∥ A^{- 1} z ∥_{2} \leq ∥ A^{- 1} ∥_{2} ∥ z ∥_{2}$ .

[F4]

If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz: a continuously differentiable vector curve has endpoint difference equal to the integral of its derivative, and hence a derivative bound $M$ gives endpoint variation at most $M$ times the interval length.

[F5]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : derivatives of a composition are obtained by composing the total derivatives.

[F6]

Finite-precision one-stage Rosenbrock step with residual status: the stage equation is $(I - γ h \tilde{J}) k = f (t, y)$ and the update is $y + h k$ .

[F7]

A computed shifted-system residual has a certified upper bound: a successful solve can supply a certified upper bound for the exact linear-equation residual.

[F8]

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: since $f$ is componentwise $C^{2}$ , all its first partial derivatives exist and are continuous; applying this theorem at each point makes $f$ totally differentiable there.

[F9]

A total derivative computes every directional derivative, and its matrix is the Jacobian: in the joint variable $(t, y)$ , the nonzero standard time and state basis directions identify the corresponding columns of $D f$ with $f_{t}$ and $D_{y} f$ .

[F10]

For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable: for an integrable vector curve, the norm of its integral is at most the integral of its Euclidean norm.

[F11]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion and The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: continuous scalar components on a compact interval are integrable, and vector integrability is componentwise.

Proof

Given: The data in the Statement.

1.1givenF4F5F8F9F10F11algebra

The componentwise $C^{2}$ hypothesis and [F8] make $f$ totally differentiable along the solution; [F9] identifies the time and state derivative blocks used in the chain rule [F5], giving $y^{''} (s) = f_{t} (s, y (s)) + D_{y} f (s, y (s)) f (s, y (s))$ . Thus $y^{'}$ and $y^{''}$ are continuous and, by [F11], integrable on the compact interval. For each $0 \leq r \leq h$ , the vector FTC [F4] applied to $y^{'}$ gives $∥ y^{'} (t + r) - y^{'} (t) ∥_{2} \leq M_{1} r$ (the case $r = 0$ is equality). Applying [F4] to $y$ then yields $y (t + h) - y (t) - h y^{'} (t) = \int_{0}^{h} (y^{'} (t + r) - y^{'} (t)) d r$ ; [F10] bounds its norm by $\int_{0}^{h} M_{1} r d r = M_{1} h^{2} / 2$ .

1.2givenF1F3F6F7algebra

If $r = \tilde{A} \hat{k} - f_{0}$ , subtraction of the exact stage equation gives $\hat{k} - k = {\tilde{A}}^{- 1} r$ . Thus [F3] and the residual bound yield $∥ \hat{k} - k ∥_{2} \leq κ_{1} ρ$ , and multiplication by $h$ bounds the update difference by $κ_{1} h ρ$ .

2.1givenF1F3step 1.1algebra

Put $f_{0} = f (t, y (t))$ . From [F1], $Φ_{γ, h}^{J} (t, y (t)) - y (t) - h f_{0} = γ h^{2} A^{- 1} J f_{0}$ . By [F3] and the bounds on $A^{- 1}, J, f_{0}$ , its norm is at most $γ κ_{0} L F h^{2}$ . Combining with step 1.1 bounds the exact-J map's local defect by $(γ κ_{0} L F + M_{1} / 2) h^{2}$ .

3.1step 2.1F2givenalgebra

Apply [F2] and $∥ f_{0} ∥_{2} \leq F$ to compare the approximate-J map with the exact-J map. The resulting difference is at most $γ h^{2} κ_{0} κ_{1} δ F$ . Adding the step 2.1 defect gives $∥ Φ_{γ, h}^{\tilde{J}} (t, y (t)) - y (t + h) ∥_{2} \leq C_{E} h^{2}$ .

4.1step 3.1step 1.2algebra∎

Add the two bounds from steps 3.1 and 1.2 by the triangle inequality. This is the claimed estimate, with no conversion of $ρ$ to forward error beyond the stated inverse bound.

Depends on

Used by

Rosenbrock–Euler has a first-order global error bound on a common tube Theorem

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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)