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Given an explicit method: $$ x_{i+1} = x_i+ h \Phi(t_i,x_i,h) $$ as predictor method and an implicit method: $$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1},h) $$ as corrector method, it follows that $$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1}^*, h), \quad (x^*_{i+1} = x_i+\Phi(t_i,x_i,h)) $$ is an explicit predictor-corrector method. If $\tau_p(h)$, $\tau_c(h)$ and $\tau_{pc}(h)$ denote respectively the local truncation error of predictor, corrector and predictor-corrector method, then $$ \tau_{pc}(h) = \tau_c(h) + \mathcal O(h \tau_p(h)) $$ (which is what I haven't proved).

From the definition of local truncation error, we have: $$ \begin{array}{rlrl} x^{(*)}(t_{i+1}) &= x(t_i) + h \Phi(t_i, x(t_i), h) + h \tau_p(t_i,h),& \tau_p(h) &= \max_i \tau_p(t_i,h)\\ x(t_{i+1}) &= x(t_i) + h \Psi(t_i, x(t_i), x(t_{i}+h), h) + h \tau_c(t_i,h),& \tau_c(h) &= \max_i \tau_c(t_i,h)\\ x(t_{i+1}) &= x(t_i) + h \Psi(t_i, x(t_i), x^*(t_i+h), h) + h \tau_{pc}(t_i,h),& \tau_{pc}(h) &= \max_i \tau_{pc}(t_i,h)\\ \end{array} $$ So, substracting from the third equation the second ecuation, and since $h > 0$: $$ \tau_{pc}(t_i,h) = \tau_c(t_i,h) - [ \Psi(t_i,x(t_i),x(t_{i+1}),h)-\Psi(t_i,x(t_i),x^*(t_{i+1}),h)] $$ And only is needed to prove that the last substraction is $\mathcal O(h \tau_p(h))$. So, $$ \|\Psi(t_i,x(t_i),x(t_{i+1}),h)-\Psi(t_i,x(t_i),x^*(t_{i+1}),h)\|\leq L\|x(t_{i+1})-x^*(t_{i+1})\| $$ $$ \leq L\|x(t_{i+1})-x(t_i)-h \Phi(t_i,x(t_i),h)-h\tau_p(t_i,h)\|\leq L\|...\| + Lh\|\tau_p(h)\| $$ where $L$ is a Lipschitz constant. How could I justify this? Any hint? Thanks in advance.

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  • $\begingroup$ The problem with you proof is that (by definition of the predictor) $$ x^*(t_{i+1}) = x(t_i) + h \Phi(t_i, x(t_i), h) $$ there's no $\tau_p$ term. There is one when $$ x(t_{i+1}) = x(t_i) + h \Phi(t_i, x(t_i), h) + h \tau_p(t_i, h) $$ $\endgroup$ – uranix Aug 16 '15 at 9:47
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Pure predictor and pure corrector schemes $$ \begin{array}{rlrlc} x(t_{i+1}) &= x(t_i) + h \Phi(t_i, x(t_i), h) + h \tau_p(t_i,h),& \tau_p(h) &= \max_i |\tau_p(t_i,h)| & (1)\\ x(t_{i+1}) &= x(t_i) + h \Psi(t_i, x(t_i), x(t_{i+1}), h) + h \tau_c(t_i,h),& \tau_c(h) &= \max_i |\tau_c(t_i,h)| & (2) \end{array} $$ And the predictor-corrector scheme $$ \begin{array}{rlrlc} x^* &= x(t_i) + h \Phi(t_i, x(t_i), h) & & & (3)\\ x(t_{i+1}) &= x(t_i) + h \Psi(t_i, x(t_i), x^*, h) + h \tau_{pc}(t_i,h),& \tau_{pc}(h) &= \max_i |\tau_{pc}(t_i,h)| & (4) \end{array} $$ Subtracting $(2)$ from $(4)$: $$ \tau_{pc}(t_i, h) = \tau_{c}(t_i, h) + \Psi(t_i, x(t_i), x(t_{i+1}), h) - \Psi(t_i, x(t_i), x^*, h) $$ Thus $$ \tau_{pc}(h) \leq \tau_{c}(h) + L ||x(t_{i+1}) - x^*|| $$ Now subtract $(1)$ from $(3)$ and get the desired result.

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