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Theory and Modern Applications

Table 1 Multicolored rooted trees for (3.2a)–(3.2b) with order less than or equal to 1.5

From: Continuous stage stochastic Runge–Kutta methods

No.

t

ρ(t)

ϕ(t)

φ(t)

1

\(\bullet _{i}\)

0.5

\(W_{i}(h)\)

\(\int _{0}^{1}z_{\tau}^{(i)}\,\mathrm{d}\tau \)

2

1

\(W_{0}(h)=h\)

\(\int _{0}^{1}z_{\tau}^{(0)}\,\mathrm{d}\tau \)

3

1

\(\int _{0}^{h}W_{j}(s)\circ \mathrm{d}W_{i}(s)\)

\(\int _{0}^{1}z_{\tau}^{(i)} \int _{0}^{1}Z_{\tau ,\xi}^{(j)}\,\mathrm{d}\xi \,\mathrm{d}\tau \)

4

1.5

\(\int _{0}^{1}W_{j}(s)W_{k}(s)\circ \mathrm{d}W_{i}(s)\)

\(\int _{0}^{1}z_{\tau}^{(i)}\int _{0}^{1}Z_{\tau ,\xi}^{(j)}\,\mathrm{d}\xi \int _{0}^{1}Z_{\tau ,\varsigma}^{(k)}\,\mathrm{d}\varsigma \,\mathrm{d}\tau \)

5

1.5

\(\int _{0}^{h} \int _{0}^{s}W_{i}(s_{1})\circ \mathrm{d}W_{j}(s_{1}) \circ \mathrm{d}W_{k}(s)\)

\(\int _{0}^{1}z_{\tau}^{(i)}\int _{0}^{1}Z_{\tau ,\xi}^{(j)} \int _{0}^{1}Z_{\xi ,\varsigma}^{(k)}\,\mathrm{d}\varsigma \,\mathrm{d}\xi \,\mathrm{d}\tau \)

6

1.5

\(\int _{0}^{h}s\circ \mathrm{d}W_{i}(s)\)

\(\int _{0}^{1}z_{\tau}^{(i)}\int Z_{\tau ,\xi}^{(0)}\,\mathrm{d}\xi \,\mathrm{d}\tau \)

7

1.5

\(\int _{0}^{h}W_{i}(s)\,\mathrm{d}s\)

\(\int _{0}^{1}z_{\tau}^{(0)} \int _{0}^{1}Z_{\tau ,\xi}^{(i)}\,\mathrm{d}\xi \,\mathrm{d}\tau \)