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m v2.04b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation) |
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where <math>\tilde{r}</math> is an order <math>\alpha </math> (>'''2''') approximation to the residual '''r''' <math>(i.e., \left\vert \mathbf{r}(t_{n},
\mathbf{z}_{n};h)-\widetilde{\mathbf{r}}(t_{n},\mathbf{z}_{n};h)\right\vert
\propto h^{\alpha+1 }).</math> The HOLL discretization (4.4) [[Rate of convergence|converges]] with order <math>\alpha</math> to the solution of nonlinear ODEs, but it match the solution of the linear ODEs.
HOLL discretizations can be derived in two ways:<ref name= ":8"/><ref name= ":3"/><ref name= ":2"/> 1) (quadrature-based) by approximating the integral representation (4.2) of '''r'''; and 2) (integrator-based) by using a numerical integrator for the differential representation of '''r''' defined by
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