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Orders and scalar stability functions of Euler, midpoint, and classical RK4

Statement

Under the smoothness and compact-tube hypotheses of Sufficient explicit Runge–Kutta order conditions through order four, the forward Euler, explicit midpoint, and classical four-stage tableaus of The explicit Euler, midpoint, and classical fourth-order Runge–Kutta tableaus have uniform local orders at least one, two, and four, respectively.

For the scalar test equation $y^{'} = λ y$ , set $z = h λ$ . The corresponding stability functions, defined by $y^{+} = R (z) y$ , are $R_{E} (z) = 1 + z, R_{M} (z) = 1 + z + \frac{z^{2}}{2}, R_{R K 4} (z) = 1 + z + \frac{z^{2}}{2} + \frac{z^{3}}{6} + \frac{z^{4}}{24} .$ The absolute stability region of a listed method is ${z \in C : ∣ R (z) ∣ \leq 1}$ . On the negative real axis, the Euler and midpoint regions are exactly $[- 2, 0]$ . No maximal real interval is asserted here for RK4.

Facts & Assumptions

Given: The three tableaus, the local exact-start defect convention, the order-condition proposition, and the scalar test equation.

[F1]

The explicit Euler, midpoint, and classical fourth-order Runge–Kutta tableaus: Euler has $A = 0, b = (1), c = 0$ ; midpoint has $A_{21} = 1 / 2, b = (0, 1), c = (0, 1 / 2)$ ; classical RK4 has the displayed four-stage coefficients.

[F2]

Sufficient explicit Runge–Kutta order conditions through order four: satisfying the coefficient equations through order $p$ gives uniform local order at least $p$ under the compact-tube hypotheses.

[F3]

One-step ODE methods, local defect, uniform order, and step stability: the exact-start local defect is the method output minus the exact next state, and local order $p$ is a uniform $O (h^{p + 1})$ defect.

[F4]

Explicit Runge–Kutta tableaus, stage maps, and failure status: an explicit tableau has a strictly lower-triangular stage matrix, row-sum abscissae, and the stated weighted stage update.

Proof

1.1F1F4algebra

The row sums of each $A$ in [F1] are its displayed vector $c$ , so all three are explicit tableaux in the required convention [F4].

1.2F1F2algebra

Euler has $\sum b_{i} = 1$ . Midpoint has $\sum b_{i} = 1$ and $\sum b_{i} c_{i} = 1 / 2$ . These are the order-one and order-two equations of [F2].

1.3F1F2algebra

For classical RK4 the row-sum vector is $c = (0, 1 / 2, 1 / 2, 1)$ , and direct row multiplication gives $A c = (0, 0, 1 / 4, 1 / 2)$ , $A c^{[2]} = (0, 0, 1 / 8, 1 / 4)$ , and $A^{2} c = (0, 0, 0, 1 / 4)$ . Therefore $\sum b_{i} = 1$ , $\sum b_{i} c_{i} = 1 / 2$ , $\sum b_{i} c_{i}^{2} = 1 / 3$ , $\sum b_{i} (A c)_{i} = 1 / 6$ , $\sum b_{i} c_{i}^{3} = 1 / 4$ , $\sum b_{i} (A c^{[2]})_{i} = 1 / 12$ , $\sum b_{i} c_{i} (A c)_{i} = 1 / 8$ , and $\sum b_{i} (A^{2} c)_{i} = 1 / 24$ .

1.4F1F4algebra

Complexify the scalar test equation $y^{'} = λ y$ . Euler multiplies $y$ by $1 + z$ . Midpoint has $k_{1} = λ y$ and $k_{2} = λ (y + \frac{h}{2} k_{1})$ , so its output factor is $1 + z + \frac{1}{2} z^{2}$ . For RK4 the successive slopes are $λ y$ , $λ y (1 + z / 2)$ , $λ y (1 + z / 2 + z^{2} / 4)$ , and $λ y (1 + z + z^{2} / 2 + z^{3} / 4)$ ; weighting them by $(1 / 6, 1 / 3, 1 / 3, 1 / 6)$ gives the displayed fourth-degree polynomial.

2.1F2step 1.2step 1.3

Apply [F2] to the sums in steps 1.2 and 1.3. It gives the asserted local orders for Euler, midpoint, and classical RK4 under exactly the hypotheses of that proposition.

2.2step 1.4algebra

For real $z$ , $∣ 1 + z ∣ \leq 1$ is equivalent to $- 2 \leq z \leq 0$ . Also $R_{M} (z) = \frac{1}{2} ((z + 1)^{2} + 1) > 0$ , so $∣ R_{M} (z) ∣ \leq 1$ is equivalent to $(z + 1)^{2} \leq 1$ , again $z \in [- 2, 0]$ . This proves the two stated scalar intervals.

3.1F2F3step 1.4step 2.2∎

The local-order statements concern exact-start defects; the stability functions concern boundedness on the scalar linear test equation. The finite intervals just computed give the explicit stability restriction $h ρ \leq 2$ for $y^{'} = - ρ y$ with $ρ > 0$ ; they assert neither nonlinear contractivity nor a matrix bound for nonnormal systems.

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Sources

Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)
MIT OpenCourseWare, 18.330 Introduction to Numerical Analysis, Chapter 5: Methods for Ordinary Differential Equations (standard reference, not scraped)