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 | |
| 32 | which 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 | |
| 36 | and 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 | |
| 40 | and 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 | |
| 44 | and 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 ] $ |