Changes between Version 1 and Version 2 of u/johannjc/scratchpad24


Ignore:
Timestamp:
03/21/17 14:49:32 (8 years ago)
Author:
Jonathan
Comment:

Legend:

Unmodified
Added
Removed
Modified
  • u/johannjc/scratchpad24

    v1 v2  
    2424We can use the midpoint rule to estimate
    2525
    26 $Q_{L_{i+1/2}} =  \displaystyle \frac{\int_{0}^{\Delta t} q_L(t) dt}{\Delta t} \approx q_L \left ( \frac{\Delta t}{2} \right )$
     26$Q_{L_{i+1/2}} =  \displaystyle \frac{\int_{0}^{\Delta t} q_L(t) dt}{\Delta t} \approx q_L \left ( \frac{\Delta t}{2} \right ) + O(\Delta t^3)$
    2727
    2828Usually some form of spatial reconstruction is used to calculate $q_L(0)$ and then the evolution equation for q is used to approximate time derivatives using spatial derivatives.
    2929
    30 $q_L(t) = q_L(0) + \displaystyle \int_0^{t} \frac{\partial q_L(t')}{\partial t} = q_L(0) + \displaystyle \int_0^{t} \left [ -\frac{\partial f(q_L(t'))}{\partial x} + s(q_L(t')) \right ]$
     30$q_L(\Delta t/2) = q(x_L,0) + \displaystyle \int_0^{t/2} \frac{\partial q(x_L,t')}{\partial t} $
     31
     32which we then use the evolution equation to replace time derivatives with spatial ones
     33
     34$q_L(\Delta t/2) = q_L(x_L,0) + \displaystyle \int_0^{t/2} \left [ -\frac{\partial f(x_L,t'))}{\partial x} + s(q(x_L,t')) \right ]$
     35
     36and then we can linearize the flux function about $q$
     37
     38$\frac{\partial f(q(x,t))}{\partial x} = \frac{\partial f}{\partial q} \frac{\partial q(x,t)}{\partial x} = A \frac{\partial q}{\partial x}$
     39
     40and use the characteristics (eigenvectors of the Jacobian of the flux) to linearize the flux
     41
     42$q_L(\Delta t/2) =  q_L(0) + \displaystyle \int_0^{t/2} \left [ A \frac{\partial (q(x_L,t'))}{\partial x} + s(q(x_L,t')) \right ] $
     43
     44and then we can use the
     45
     46$q_L(\Delta t/2) =  q_L(0) + A \displaystyle \int_0^{t/2} \left [ \frac{\partial (q())}{\partial x} + s(q_L(t')) \right ] $