Lines Matching refs:t
21 \frac{\partial \rho}{\partial t} + \nabla \cdot \bm{U} &= 0 \\
22 \frac{\partial \bm{U}}{\partial t} + \nabla \cdot \left( \frac{\bm{U} \otimes \bm{U}}{\rho} + P \bm…
23 \frac{\partial E}{\partial t} + \nabla \cdot \left( \frac{(E + P)\bm{U}}{\rho} -\bm{u} \cdot \bm{\s…
39 \frac{\partial \bm{q}}{\partial t} + \nabla \cdot \bm{F}(\bm{q}) -S(\bm{q}) = 0 \, ,
77 \bm{q}_N (\bm{x},t)^{(e)} = \sum_{k=1}^{P}\psi_k (\bm{x})\bm{q}_k^{(e)}
86 \int_{\Omega} \bm v \cdot \left(\frac{\partial \bm{q}_N}{\partial t} + \nabla \cdot \bm{F}(\bm{q}_N…
95 \int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) \…
116 \bm{q}_N^{n+1} = \bm{q}_N^n + \Delta t \sum_{i=1}^{s} b_i k_i \, ,
123 k_1 &= f(t^n, \bm{q}_N^n)\\
124 k_2 &= f(t^n + c_2 \Delta t, \bm{q}_N^n + \Delta t (a_{21} k_1))\\
125 k_3 &= f(t^n + c_3 \Delta t, \bm{q}_N^n + \Delta t (a_{31} k_1 + a_{32} k_2))\\
127 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)\\
134 f(t^n, \bm{q}_N^n) = - [\nabla \cdot \bm{F}(\bm{q}_N)]^n + [S(\bm{q}_N)]^n \, .
143 \bm f(\bm q_N) \equiv \bm g(t^{n+1}, \bm{q}_N, \bm{\dot{q}}_N) = 0 \, ,
160 …The scalar "shift" $\alpha$ scales inversely with the time step $\Delta t$, so small time steps re…
179 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
182 … F_{\text{adv}}}{\partial \bm q}\right) \bm\tau \left( \frac{\partial \bm{q}_N}{\partial t} \, + \,
196 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
273 \mathcal{F} = \sqrt{ \rho^2 \left [ \left(\frac{2C_t}{\Delta t}\right)^2
284 \tau = \left [ \left(\frac{2 C_t}{\Delta t}\right)^2
328 …= \frac{1}{L_z + (T_f - T_0)}\int_0^{L_z} \int_{T_0}^{T_f} \phi(x, y, z, t) \mathrm{d}t \mathrm{d}z
349 \langle \phi \rangle_z(x,y,t) = \frac{1}{L_z} \int_0^{L_z} \phi(x, y, z, t) \mathrm{d}z
362 …_N = \int_0^{L_x} \int_0^{L_y} \left [\frac{1}{L_z} \int_0^{L_z} \phi(x,y,z,t) \mathrm{d}z \right …
368 \bm M [\langle \phi \rangle_z]_N = \frac{1}{L_z} \int_\Omega \phi(x,y,z,t) \psi^\mathrm{parent}_N(x…
388 …_z + (T_f - T_0)} \int_\Omega \int_{T_0}^{T_f} \phi(x,y,z,t) \psi^\mathrm{parent}_N \mathrm{d}t \m…
390 where the integral $\int_{T_0}^{T_f} \phi(x,y,z,t) \mathrm{d}t$ is calculated on a running basis.
525 \frac{\partial E}{\partial t} + \nabla \cdot (\bm{u} E ) - \kappa \nabla E = 0 \, ,
561 \frac{\partial \rho}{\partial t} + \nabla \cdot \bm{U} &= 0 \\
562 \frac{\partial \bm{U}}{\partial t} + \nabla \cdot \left( \frac{\bm{U} \otimes \bm{U}}{\rho} + P \bm…
563 \frac{\partial E}{\partial t} + \nabla \cdot \left( \frac{(E + P)\bm{U}}{\rho} \right) &= 0 \, , \\
641 At time $t=0$, this domain is subjected to freestream boundary conditions at the inflow (left) and …
664 …defined in terms of the Exner pressure, $\pi(\bm{x},t)$, and potential temperature, $\theta(\bm{x}…
667 …{P_0}{( c_p - c_v)\theta(\bm{x},t)} \pi(\bm{x},t)^{\frac{c_v}{ c_p - c_v}} \, , \\ e &= c_v \theta…
721 \bm{u}(\bm{x}, t) = \bm{\overline{u}}(\bm{x}) + \bm{C}(\bm{x}) \cdot \bm{v}'
726 …qrt{q^n(\bm{x})} \bm{\sigma}^n \cos(\kappa^n \bm{d}^n \cdot \bm{\hat{x}}^n(\bm{x}, t) + \phi^n ) \\
727 \bm{\hat{x}}^n &= \left[(x - U_0 t)\max(2\kappa_{\min}/\kappa^n, 0.1) , y, z \right]^T