Changeset 9407 for branches/2017/dev_merge_2017/DOC/tex_sub/annex_E.tex
- Timestamp:
- 2018-03-15T17:40:35+01:00 (6 years ago)
- File:
-
- 1 edited
Legend:
- Unmodified
- Added
- Removed
-
branches/2017/dev_merge_2017/DOC/tex_sub/annex_E.tex
r9393 r9407 5 5 % ================================================================ 6 6 \chapter{Note on some algorithms} 7 \label{ Apdx_E}7 \label{apdx:E} 8 8 \minitoc 9 9 … … 20 20 % ------------------------------------------------------------------------------------------------------------- 21 21 \section{Upstream Biased Scheme (UBS) (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})} 22 \label{ TRA_adv_ubs}22 \label{sec:TRA_adv_ubs} 23 23 24 24 The UBS advection scheme is an upstream biased third order scheme based on … … 26 26 QUICK scheme (Quadratic Upstream Interpolation for Convective 27 27 Kinematics). For example, in the $i$-direction : 28 \begin{equation} \label{ Eq_tra_adv_ubs2}28 \begin{equation} \label{eq:tra_adv_ubs2} 29 29 \tau _u^{ubs} = \left\{ \begin{aligned} 30 30 & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ … … 33 33 \end{equation} 34 34 or equivalently, the advective flux is 35 \begin{equation} \label{ Eq_tra_adv_ubs2}35 \begin{equation} \label{eq:tra_adv_ubs2} 36 36 U_{i+1/2} \ \tau _u^{ubs} 37 37 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} … … 61 61 scheme when \np{ln\_traadv\_ubs}\forcode{ = .true.}. 62 62 63 For stability reasons, in \ eqref{Eq_tra_adv_ubs}, the first term which corresponds63 For stability reasons, in \autoref{eq:tra_adv_ubs}, the first term which corresponds 64 64 to a second order centred scheme is evaluated using the \textit{now} velocity 65 65 (centred in time) while the second term which is the diffusive part of the scheme, … … 67 67 by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK 68 68 schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 69 (\ ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.69 (\autoref{eq:tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 70 70 This option is not available through a namelist parameter, since the 1/6 71 71 coefficient is hard coded. Nevertheless it is quite easy to make the … … 87 87 eight-order accurate conventional scheme. 88 88 89 NB 3 : It is straight forward to rewrite \ eqref{Eq_tra_adv_ubs} as follows:90 \begin{equation} \label{ Eq_tra_adv_ubs2}89 NB 3 : It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 90 \begin{equation} \label{eq:tra_adv_ubs2} 91 91 \tau _u^{ubs} = \left\{ \begin{aligned} 92 92 & \tau _u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ … … 95 95 \end{equation} 96 96 or equivalently 97 \begin{equation} \label{ Eq_tra_adv_ubs2}97 \begin{equation} \label{eq:tra_adv_ubs2} 98 98 \begin{split} 99 99 e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} … … 102 102 \end{split} 103 103 \end{equation} 104 \ eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that104 \autoref{eq:tra_adv_ubs2} has several advantages. First it clearly evidence that 105 105 the UBS scheme is based on the fourth order scheme to which is added an 106 106 upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order 107 107 part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order 108 part as stated above using \ eqref{Eq_tra_adv_ubs}. Third, the diffusive term is108 part as stated above using \autoref{eq:tra_adv_ubs}. Third, the diffusive term is 109 109 in fact a biharmonic operator with a eddy coefficient with is simply proportional 110 110 to the velocity. 111 111 112 112 laplacian diffusion: 113 \begin{equation} \label{ Eq_tra_ldf_lap}113 \begin{equation} \label{eq:tra_ldf_lap} 114 114 \begin{split} 115 115 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i … … 124 124 125 125 bilaplacian: 126 \begin{equation} \label{ Eq_tra_ldf_lap}126 \begin{equation} \label{eq:tra_ldf_lap} 127 127 \begin{split} 128 128 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 136 136 $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 137 137 it comes : 138 \begin{equation} \label{ Eq_tra_ldf_lap}138 \begin{equation} \label{eq:tra_ldf_lap} 139 139 \begin{split} 140 140 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\; e_{3T}} \\ … … 146 146 \end{equation} 147 147 if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is 148 \begin{equation} \label{ Eq_tra_ldf_lap}148 \begin{equation} \label{eq:tra_ldf_lap} 149 149 \begin{split} 150 150 F_u^{lT} = - \frac{1}{12} … … 157 157 beurk.... reverte the logic: starting from the diffusive part of the advective flux it comes: 158 158 159 \begin{equation} \label{ Eq_tra_adv_ubs2}159 \begin{equation} \label{eq:tra_adv_ubs2} 160 160 \begin{split} 161 161 F_u^{lT} … … 166 166 167 167 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 168 \begin{equation} \label{ Eq_tra_adv_ubs2}168 \begin{equation} \label{eq:tra_adv_ubs2} 169 169 \begin{split} 170 170 F_u^{lT} … … 175 175 176 176 sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 177 \begin{equation} \label{ Eq_tra_adv_ubs2}177 \begin{equation} \label{eq:tra_adv_ubs2} 178 178 \begin{split} 179 179 F_u^{lT} … … 189 189 % ------------------------------------------------------------------------------------------------------------- 190 190 \section{Leapfrog energetic} 191 \label{ LF}191 \label{sec:LF} 192 192 193 193 We adopt the following semi-discrete notation for time derivative. Given the values of a variable $q$ at successive time step, the time derivation and averaging operators at the mid time step are: 194 \begin{subequations} \label{ dt_mt}194 \begin{subequations} \label{eq:dt_mt} 195 195 \begin{align} 196 196 \delta _{t+\rdt/2} [q] &= \ \ \, q^{t+\rdt} - q^{t} \\ … … 202 202 , respectively. 203 203 204 The Leap-frog time stepping given by \ eqref{Eq_DOM_nxt} can be defined as:205 \begin{equation} \label{ LF}204 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 205 \begin{equation} \label{eq:LF} 206 206 \frac{\partial q}{\partial t} 207 207 \equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} 208 208 = \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 209 209 \end{equation} 210 Note that \ eqref{LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$210 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$ 211 211 as it can be found sometime in literature. 212 212 The leap-Frog time stepping is a second order centered scheme. As such it respects 213 213 the quadratic invariant in integral forms, $i.e.$ the following continuous property, 214 \begin{equation} \label{ Energy}214 \begin{equation} \label{eq:Energy} 215 215 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 216 216 =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} … … 252 252 scheme, but is formulated within the \NEMO framework ($i.e.$ using scale 253 253 factors rather than grid-size and having a position of $T$-points that is not 254 necessary in the middle of vertical velocity points, see Fig.~\ref{Fig_zgr_e3}).255 256 In the formulation \ eqref{Eq_tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO,254 necessary in the middle of vertical velocity points, see \autoref{fig:zgr_e3}). 255 256 In the formulation \autoref{eq:tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 257 257 the off-diagonal terms of the small angle diffusion tensor contain several double 258 258 spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. … … 263 263 In other word, the operator applied to a tracer does not warranties the decrease of 264 264 its global average variance. To circumvent this, we have introduced a smoothing of 265 the slopes of the iso-neutral surfaces (see \ S\ref{LDF}). Nevertheless, this technique265 the slopes of the iso-neutral surfaces (see \autoref{chap:LDF}). Nevertheless, this technique 266 266 works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation 267 267 of density), but it does not work for a passive tracer. \citep{Griffies_al_JPO98} introduce … … 270 270 with a derivative in the same direction by considering triads. For example in the 271 271 (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 272 \begin{equation} \label{ Gf_triads}272 \begin{equation} \label{eq:Gf_triads} 273 273 _i^k \mathbb{T}_{i_p}^{k_p} (T) 274 274 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k} \ A_i^k \left( … … 282 282 $A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, 283 283 and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad : 284 \begin{equation} \label{ Gf_slopes}284 \begin{equation} \label{eq:Gf_slopes} 285 285 _i^k \mathbb{R}_{i_p}^{k_p} 286 286 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac … … 288 288 {\left(\alpha / \beta \right)_i^k \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 289 289 \end{equation} 290 Note that in \ eqref{Gf_slopes} we use the ratio $\alpha / \beta$ instead of290 Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of 291 291 multiplying the temperature derivative by $\alpha$ and the salinity derivative 292 292 by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be 293 293 evaluated directly. 294 294 295 Note that in \ eqref{Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of295 Note that in \autoref{eq:Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of 296 296 ${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease 297 297 of tracer variance and the presence of partial cell at the ocean bottom 298 (see Appendix~\ref{Apdx_Gf_operator}).298 (see \autoref{apdx:Gf_operator}). 299 299 300 300 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 301 \begin{figure}[!ht] \label{Fig_ISO_triad} 302 \begin{center} 301 \begin{figure}[!ht] \begin{center} 303 302 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 304 \caption{ \protect\label{ Fig_ISO_triad}303 \caption{ \protect\label{fig:ISO_triad} 305 304 Triads used in the Griffies's like iso-neutral diffision scheme for 306 305 $u$-component (upper panel) and $w$-component (lower panel).} … … 311 310 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 312 311 They take the following expression : 313 \begin{flalign} \label{ Gf_fluxes}312 \begin{flalign} \label{eq:Gf_fluxes} 314 313 \begin{split} 315 314 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) … … 322 321 323 322 The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the 324 sum of the fluxes that cross the $u$- and $w$-face ( Fig.~\ref{Fig_ISO_triad}):325 \begin{flalign} \label{ Eq_iso_flux}323 sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): 324 \begin{flalign} \label{eq:iso_flux} 326 325 \textbf{F}_{iso}(T) 327 326 &\equiv \sum_{\substack{i_p,\,k_p}} … … 353 352 resulting in a iso-neutral diffusion tendency on temperature given by the divergence 354 353 of the sum of all the four triad fluxes : 355 \begin{equation} \label{ Gf_operator}354 \begin{equation} \label{eq:Gf_operator} 356 355 D_l^T = \frac{1}{b_T} \sum_{\substack{i_p,\,k_p}} \left\{ 357 356 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] … … 365 364 \item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator 366 365 recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction : 367 \begin{equation} \label{ Gf_property1a}366 \begin{equation} \label{eq:Gf_property1a} 368 367 D_l^T = \frac{1}{b_T} \ \delta_{i} 369 368 \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] … … 388 387 \item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of locally referenced 389 388 potential density is zero, $i.e.$ 390 \begin{align} \label{ Gf_property2}389 \begin{align} \label{eq:Gf_property2} 391 390 \begin{matrix} 392 391 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} … … 398 397 \end{matrix} 399 398 \end{align} 400 This result is trivially obtained using the \ eqref{Gf_triads} applied to $T$ and $S$401 and the definition of the triads' slopes \ eqref{Gf_slopes}.399 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ 400 and the definition of the triads' slopes \autoref{eq:Gf_slopes}. 402 401 403 402 \item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the 404 403 total tracer content, $i.e.$ 405 \begin{equation} \label{ Gf_property1}404 \begin{equation} \label{eq:Gf_property1} 406 405 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 407 406 \end{equation} … … 411 410 \item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does 412 411 not increase the total tracer variance, $i.e.$ 413 \begin{equation} \label{ Gf_property1}412 \begin{equation} \label{eq:Gf_property1} 414 413 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 415 414 \end{equation} 416 The property is demonstrated in the Appendix~\ref{Apdx_Gf_operator}. It is a415 The property is demonstrated in the \autoref{apdx:Gf_operator}. It is a 417 416 key property for a diffusion term. It means that the operator is also a dissipation 418 417 term, $i.e.$ it is a sink term for the square of the quantity on which it is applied. … … 422 421 \item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, 423 422 $i.e.$ 424 \begin{equation} \label{ Gf_property1}423 \begin{equation} \label{eq:Gf_property1} 425 424 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 426 425 \end{equation} … … 428 427 operator. We just have to apply the same routine. This properties can be demonstrated 429 428 quite easily in a similar way the "non increase of tracer variance" property has been 430 proved (see Appendix~\ref{Apdx_Gf_operator}).429 proved (see \autoref{apdx:Gf_operator}). 431 430 \end{description} 432 431 … … 442 441 eddy induced velocity, the formulation of which depends on the slopes of iso- 443 442 neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used 444 here are referenced to the geopotential surfaces, $i.e.$ \ eqref{Eq_ldfslp_geo}445 is used in $z$-coordinate, and the sum \ eqref{Eq_ldfslp_geo}446 + \ eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates.443 here are referenced to the geopotential surfaces, $i.e.$ \autoref{eq:ldfslp_geo} 444 is used in $z$-coordinate, and the sum \autoref{eq:ldfslp_geo} 445 + \autoref{eq:ldfslp_iso} in $z^*$ or $s$-coordinates. 447 446 448 447 The eddy induced velocity is given by: 449 \begin{equation} \label{ Eq_eiv_v}448 \begin{equation} \label{eq:eiv_v} 450 449 \begin{split} 451 450 u^* & = - \frac{1}{e_2\,e_{3}} \;\partial_k \left( e_2 \, A_e \; r_i \right) … … 467 466 A traditional way to implement this additional advection is to add it to the eulerian 468 467 velocity prior to compute the tracer advection. This allows us to take advantage of 469 all the advection schemes offered for the tracers (see \ S\ref{TRA_adv}) and not just468 all the advection schemes offered for the tracers (see \autoref{sec:TRA_adv}) and not just 470 469 a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers 471 470 where \emph{positivity} of the advection scheme is of paramount importance. 472 % give here the expression using the triads. It is different from the one given in \ eqref{Eq_ldfeiv}471 % give here the expression using the triads. It is different from the one given in \autoref{eq:ldfeiv} 473 472 % see just below a copy of this equation: 474 %\begin{equation} \label{ Eq_ldfeiv}473 %\begin{equation} \label{eq:ldfeiv} 475 474 %\begin{split} 476 475 % u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ … … 479 478 %\end{split} 480 479 %\end{equation} 481 \begin{equation} \label{ Eq_eiv_vd}480 \begin{equation} \label{eq:eiv_vd} 482 481 \textbf{F}_{eiv}^T \equiv \left( \begin{aligned} 483 482 \sum_{\substack{i_p,\,k_p}} & … … 491 490 \end{equation} 492 491 493 \ ref{Griffies_JPO98} introduces another way to implement the eddy induced advection,492 \citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, 494 493 the so-called skew form. It is based on a transformation of the advective fluxes 495 494 using the non-divergent nature of the eddy induced velocity. … … 522 521 and since the eddy induces velocity field is no-divergent, we end up with the skew 523 522 form of the eddy induced advective fluxes: 524 \begin{equation} \label{ Eq_eiv_skew_continuous}523 \begin{equation} \label{eq:eiv_skew_continuous} 525 524 \textbf{F}_{eiv}^T = \begin{pmatrix} 526 525 {+ e_{2} \, A_{e} \; r_i \; \partial_k T} \\ … … 529 528 \end{equation} 530 529 The tendency associated with eddy induced velocity is then simply the divergence 531 of the \ eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer530 of the \autoref{eq:eiv_skew_continuous} fluxes. It naturally conserves the tracer 532 531 content, as it is expressed in flux form and, as the advective form, it preserve the 533 tracer variance. Another interesting property of \ eqref{Eq_eiv_skew_continuous}532 tracer variance. Another interesting property of \autoref{eq:eiv_skew_continuous} 534 533 form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral 535 534 diffusion and eddy induced velocity terms: 536 \begin{flalign} \label{ Eq_eiv_skew+eiv_continuous}535 \begin{flalign} \label{eq:eiv_skew+eiv_continuous} 537 536 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 538 537 \begin{pmatrix} … … 554 553 has been used to reduce the computational time \citep{Griffies_JPO98}, but it is 555 554 not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to 556 choose a discret form of \ eqref{Eq_eiv_skew_continuous} which is consistent with the557 iso-neutral operator \ eqref{Gf_operator}. Using the slopes \eqref{Gf_slopes}555 choose a discret form of \autoref{eq:eiv_skew_continuous} which is consistent with the 556 iso-neutral operator \autoref{eq:Gf_operator}. Using the slopes \autoref{eq:Gf_slopes} 558 557 and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient), 559 558 the resulting discret form is given by: 560 \begin{equation} \label{ Eq_eiv_skew}559 \begin{equation} \label{eq:eiv_skew} 561 560 \textbf{F}_{eiv}^T \equiv \frac{1}{4} \left( \begin{aligned} 562 561 \sum_{\substack{i_p,\,k_p}} & … … 569 568 \end{aligned} \right) 570 569 \end{equation} 571 Note that \ eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells.570 Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 572 571 In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces 573 572 must be added to $\mathbb{R}$ for the discret form to be exact. … … 575 574 Such a choice of discretisation is consistent with the iso-neutral operator as it uses the 576 575 same definition for the slopes. It also ensures the conservation of the tracer variance 577 (see Appendix \ ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component576 (see Appendix \autoref{apdx:eiv_skew}), $i.e.$ it does not include a diffusive component 578 577 but is a "pure" advection term. 579 578 … … 586 585 % ================================================================ 587 586 \subsection{Discrete invariants of the iso-neutral diffrusion} 588 \label{ Apdx_Gf_operator}587 \label{subsec:Gf_operator} 589 588 590 589 Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. … … 596 595 \int_D D_l^T \; T \;dv \leq 0 597 596 \end{align*} 598 The discrete form of its left hand side is obtained using \ eqref{Eq_iso_flux}597 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 599 598 600 599 \begin{align*} … … 673 672 % 674 673 \allowdisplaybreaks 675 \intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \ eqref{Gf_triads}. It becomes: }674 \intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \autoref{eq:Gf_triads}. It becomes: } 676 675 % 677 676 &\equiv -\sum_{i,k} … … 739 738 % ================================================================ 740 739 \subsection{Discrete invariants of the skew flux formulation} 741 \label{ Apdx_eiv_skew}740 \label{subsec:eiv_skew} 742 741 743 742 … … 750 749 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv \equiv 0 751 750 \end{align*} 752 The discrete form of its left hand side is obtained using \ eqref{Eq_eiv_skew}751 The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} 753 752 \begin{align*} 754 753 \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}} \Biggl\{ \;\;
Note: See TracChangeset
for help on using the changeset viewer.
