Lines Matching refs:q
31 \frac{\partial \bm{q}}{\partial t} + \nabla \cdot \bm{F}(\bm{q}) -S(\bm{q}) = 0 \, ,
37 \bm{q} =
54 \bm{F}(\bm{q}) &=
65 S(\bm{q}) &=
79 \bm{q}_N (\bm{x},t)^{(e)} = \sum_{k=1}^{P}\psi_k (\bm{x})\bm{q}_k^{(e)}
88 …Omega} \bm v \cdot \left(\frac{\partial \bm{q}_N}{\partial t} + \nabla \cdot \bm{F}(\bm{q}_N) - \b…
97 \int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) \…
98 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
120 \bm{q}_N^{n+1} = \bm{q}_N^n + \Delta t \sum_{i=1}^{s} b_i k_i \, ,
127 k_1 &= f(t^n, \bm{q}_N^n)\\
128 k_2 &= f(t^n + c_2 \Delta t, \bm{q}_N^n + \Delta t (a_{21} k_1))\\
129 k_3 &= f(t^n + c_3 \Delta t, \bm{q}_N^n + \Delta t (a_{31} k_1 + a_{32} k_2))\\
131 k_i &= f\left(t^n + c_i \Delta t, \bm{q}_N^n + \Delta t \sum_{j=1}^s a_{ij} k_j \right)\\
138 f(t^n, \bm{q}_N^n) = - [\nabla \cdot \bm{F}(\bm{q}_N)]^n + [S(\bm{q}_N)]^n \, .
147 \bm f(\bm q_N) \equiv \bm g(t^{n+1}, \bm{q}_N, \bm{\dot{q}}_N) = 0 \, ,
150 where the time derivative $\bm{\dot q}_N$ is defined by
153 \bm{\dot{q}}_N(\bm q_N) = \alpha \bm q_N + \bm z_N
161 …} = \frac{\partial \bm g}{\partial \bm q_N} + \alpha \frac{\partial \bm g}{\partial \bm{\dot q}_N}.
183 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
184 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
185 + \int_{\partial \Omega} \bm v \cdot \bm{F}(\bm{q}_N) \cdot \widehat{\bm{n}} \,dS & \\
186 …\left(\frac{\partial \bm F_{\text{adv}}}{\partial \bm q}\right) \bm\tau \left( \frac{\partial \bm{…
187 \nabla \cdot \bm{F} \, (\bm{q}_N) - \bm{S}(\bm{q}_N) \right) \,dV &= 0
200 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
201 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
202 + \int_{\partial \Omega} \bm v \cdot \bm{F}(\bm{q}_N) \cdot \widehat{\bm{n}} \,dS & \\
203 …left(\frac{\partial \bm F_{\text{adv}}}{\partial \bm q}\right) \bm\tau \nabla \cdot \bm{F} \, (\bm…
211 …xt{adv}}}{\partial \bm q}$ (rather than its transpose) can be explained via an ansatz for subgrid …
216 \diff\bm F_{\text{adv}}(\diff\bm q; \bm q) &= \frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \d…
251 For advection-diffusion, $\bm F(q) = \bm u q$, and thus the SU stabilization term is
254 …dot \bm u \tau \bm u \cdot \nabla q = \nabla_{\bm X} v \cdot (\bm u_{\bm X} \tau \bm u_{\bm X}) \c…
317 The flux Jacobian $\frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm n}_i$ in each …
329 \lambda_{\max \text{abs}} \Bigl( \frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm …
341 …-weak-vector-ns-su` features the term $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$, the divergence…
345 …perform a projection operation to get $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$ at quadrature p…
360 …ction of the divergence of the diffusive flux itself, $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$.
363 To do this, look at the RHS of the $L^2$ projection of $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$:
366 \int_{\Omega} \bm v \cdot \nabla \cdot \bm F_{\text{diff}}(\bm{q}_N) \,dV
369 As noted, we can't calculate $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$ at quadrature points, so …
373 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}_{\text{diff}}(\bm{q}_N)\,dV
377 After the projection, $\nabla \cdot \bm F_{\text{diff}}(\bm{q}_N)$ is interpolated directly to quad…
394 \frac{\partial \bm{\overline q}}{\partial t} + \nabla \cdot \bm{\overline F}(\bm{\overline q}) -S(\…
400 \bm{\overline F}(\bm{\overline q}) =
401 \bm{F} (\bm{\overline q}) +
520 \bm{v}' &= 2 \sqrt{3/2} \sum^N_{n=1} \sqrt{q^n(\bm{x})} \bm{\sigma}^n \cos(\kappa^n \bm{d}^n \cdot …
525 …(such that $\bm{R} = \bm{CC}^T$ ), bulk velocity $U_0$, wavemode amplitude $q^n$, wavemode frequen…
540 The wavemode amplitudes $q^n$ are defined by a model energy spectrum $E(\kappa)$:
543 q^n = \frac{E(\kappa^n) \Delta \kappa^n}{\sum^N_{n=1} E(\kappa^n)\Delta \kappa^n} \ ,\quad \Delta \…
611 …olation (for $l_t$, $\varepsilon$, $C_{ij}$, and $\overline{\bm u}$) or by calculation (for $q^n$).
640 | $q^n$ | q^n | Yes | Yes |
719 S(\bm{q}) = -\sigma(\bm{x})\left.\frac{\partial \bm{q}}{\partial \bm{Y}}\right\rvert_{\bm{q}} \bm{Y…
723 …\bm Y'$ converted to conservative source using $\partial \bm{q}/\partial \bm{Y}\rvert_{\bm{q}}$, w…