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b/documentation/gravity_current_umana_anderson.tex index d9ff896..784a12c 100644 --- a/documentation/gravity_current_umana_anderson.tex +++ b/documentation/gravity_current_umana_anderson.tex @@ -433,10 +433,11 @@ Note that in this final expression $dr/ds$ is also evaluated at $r=1$ (where $s= \bea \left. \frac{dr}{ds} \right|_{r=1} & = & \frac{1}{\lambda} \left( e^\lambda - 1 \right). \eea +Also note that $dr/ds \rightarrow 1$ as $\lambda \rightarrow 0$, which is the limit in which the uniform grid is recovered. -In our numerical codes {\tt gc\_molND\_nonuniform\_s.m} and {\tt gc\_rhsNC\_nonuniform\_s.m} we introduce equally-spaced points in terms of the variable $r$ which -allows relatively straightforward expression of derivatives in terms of finite differences but with the grid points in terms of $s$ (and hence $x = s x_N(t)$) more tightly -spaced near the leading edge of the gravity current. +In our numerical codes {\tt gc\_molND\_nonuniform\_s.m} and {\tt gc\_rhsND\_nonuniform\_s.m} we introduce equally-spaced points in terms of the variable $r$ which +allows relatively straightforward finite difference expressions for spatial derivatives but with the grid points in terms of $s$ (and hence $x = s x_N(t)$) more tightly +spaced near the leading edge of the gravity current with the parameter $\lambda$ that helps control the refinement. @@ -515,11 +516,11 @@ H(\zeta=0) & = & 0, \\ The equation~(\ref{eq:leading_order_inner}) can be integrated once to give \bea -\frac{dx_N}{dt} H & = & \left( \frac{1}{3} H^3 + H \right) \frac{dH}{d\eta} + c_0, +\frac{dx_N}{dt} H & = & \left( \frac{1}{3} H^3 + H \right) \frac{dH}{d\zeta} + c_0, \eea where $c_0$ is a constant. The boundary conditions require $c_0=0$. We then have \bea -\frac{dx_N}{dt} & = & \left( \frac{1}{3} H^2 + 1 \right) \frac{dH}{d\eta} = \frac{d}{d\zeta} \left[ \frac{1}{9} H^3 + H \right] +\frac{dx_N}{dt} & = & \left( \frac{1}{3} H^2 + 1 \right) \frac{dH}{d\zeta} = \frac{d}{d\zeta} \left[ \frac{1}{9} H^3 + H \right] \eea One further integration and application of boundary conditions yields \bea @@ -607,10 +608,10 @@ Note that the PDE in~(\ref{eq:outer_h_fixed_domain}) requires that we have acces expresses the equation for the evolution of the actual leading-edge location $x_N(t)$. Converting this to a domain in which non-equally-spaced points can be implemented using the same relationship between $r$ and $s$ as used earlier gives -This leads to the modified system \bea \label{eq:outer_h_fixed_domain_drds} -\frac{\partial h}{\partial t} & = & \frac{s}{x_M} \frac{dx_M}{dt} \frac{dr}{ds} \frac{\partial h}{\partial r} + \frac{1}{x_M^2} \frac{dr}{ds} \frac{\partial}{\partial r} \left[ f(h) \frac{dr}{ds} \frac{\partial h}{\partial r} \right], \hspace{0.25in} \mbox{on $0 < r < 1$},\\ +\frac{\partial h}{\partial t} & = & \frac{s}{x_M} \frac{dx_M}{dt} \frac{dr}{ds} \frac{\partial h}{\partial r} + \frac{1}{x_M^2} \frac{dr}{ds} \frac{\partial}{\partial r} +\left[ f(h) \frac{dr}{ds} \frac{\partial h}{\partial r} \right], \hspace{0.25in} \mbox{on $0 < r < 1$},\\ \label{eq:outer_bc1_fixed_domain_drds} \frac{\partial h}{\partial r}(r=0,t) & = & 0,\\ \label{eq:outer_bc2_fixed_domain_drds} -- 2.43.0