38 | | $\frac{\partial f(q(x,t))}{\partial x} = \frac{\partial f}{\partial q} \frac{\partial q(x,t)}{\partial x} = A \frac{\partial q}{\partial x}$ |
| 38 | $\frac{\partial f(q(x,t))}{\partial x} = \frac{\partial f}{\partial q} \frac{\partial q(x,t)}{\partial x} = \lambda \frac{\partial q(x,t)}{\partial x} $ |
| 39 | |
| 40 | |
| 41 | $q_L(\Delta t/2) = q_L(0) + \displaystyle \int_0^{t/2} \lambda \frac{\partial (q(x_L,t'))}{\partial x} dt'+ \int_0^{t/2} s(q(x_L,t')) dt' $ |
| 42 | |
| 43 | Now if we ignore the source term, we can directly solve for |
| 44 | |
| 45 | $q(x,t)=q(x-\lambda t, 0)$ |
| 46 | |
| 47 | and under the change of variables $x'=x_L-\lambda t'$ we can rewrite the integrals as |
| 48 | |
| 49 | $q_L(\Delta t/2) = q_L(0) + \displaystyle \int_{x_L}^{x_L-\lambda t/2} \frac{\partial (q(x',0))}{\partial x} dx' + \int_{x_L}^{x_L-\lambda t/2} s(q(x',0)) dx'$ |
| 50 | |
| 51 | which just gives us |
| 52 | |
| 53 | $q_L(\Delta t/2) = q_L(0) - (q(x_L,0) - q(x_L-\lambda \Delta t/2)) + s(\bar{q})\Delta t/2$ |
| 54 | |
| 55 | where $\bar{q} = \displaystyle \int_{x_L-\lambda \Delta t/2}^{x_L}q(x',0) dx'$ |
| 56 | |
| 57 | |
| 58 | |
| 59 | $\frac{\partial f(q(x,t))}{\partial x} = \frac{\partial f}{\partial q} \frac{\partial q(x,t)}{\partial x} = A \frac{\partial q(x,t)}{\partial x} = R \Lambda L \frac{\partial q(x,t)}{\partial x} = \displaystyle \sum_i R_i \lambda_i L_i \frac{\partial q(x,t)}{\partial x} \approx \displaystyle \sum_i R_i \lambda_i L_i \frac{\partial q(x-\lambda_i t,0)}{\partial x} $ |
| 60 | |
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 ] $ |
| 68 | $q_L(\Delta t/2) = q_L(0) + \displaystyle \int_0^{t/2} \left [ \sum R_i \lambda_i \delta l (L_i \frac{\partial (q(x_L,t'))}{\partial x} + s(q(x_L,t')) \right ] $ |
| 69 | |
| 70 | |
| 71 | $q_L(\Delta t/2) = q_L(0) + \displaystyle \int_0^{t/2} \left [ \sum R_i \lambda_i \frac{\partial (q(x_L,t'))}{\partial x} + s(q(x_L,t')) \right ] $ |
| 72 | |
| 73 | |
| 74 | |
| 75 | and then we can use the solution for the characteristics $q(x_L,t')=q(x_L-\lambda t',0)$ to turn the time integral into a spatial integral |
| 76 | |
| 77 | $q_L(\Delta t/2) = q_L(0) + A \displaystyle \int_0^{t/2} \left [ \frac{\partial (q(x_L-\lambda t',0))}{\partial x} + s(q(x_L-\lambda t',0)) \right ] $ |
| 78 | |
| 79 | which we can do a variable substitution $x' = \lambda t'$ |