1. CE 498/698 and ERS 685 (Spring 2004) Lecture 121 Lecture 12: Control-Volume Approach (time-variable) CE 498/698 and ERS 685 Principles of Water Quality Modeling

2. CE 498/698 and ERS 685 (Spring 2004) Lecture 122 t x +1 i-1i+1 i i, i+1, i, +1 i-1, i, -1 tt xx forward difference over time finite difference approximations

3. CE 498/698 and ERS 685 (Spring 2004) Lecture 123 t x +1 i-1i+1 i i, i+1, i, +1 i-1, i, -1 tt xx centered difference over space finite difference approximations

4. CE 498/698 and ERS 685 (Spring 2004) Lecture 124 t x +1 i-1i+1 i i, i+1, i, +1 i-1, i, -1 tt xx centered difference over space finite difference approximations

5. CE 498/698 and ERS 685 (Spring 2004) Lecture 125 t x +1 i-1i+1 i i, i+1, i, +1 i-1, i, -1 tt xx backward difference over space finite difference approximations

6. CE 498/698 and ERS 685 (Spring 2004) Lecture 126 finite difference approximations forward difference over time centered difference over space centered difference over space FTCS

7. CE 498/698 and ERS 685 (Spring 2004) Lecture 127 +1 i-1i+1i second derivative with spacefirst derivative with time FTCS explicit method

8. CE 498/698 and ERS 685 (Spring 2004) Lecture 128 Control-Volume Approach must be positive!

9. CE 498/698 and ERS 685 (Spring 2004) Lecture 129 Solution stability Assuming Q, E, V, k are constant: What if we used backward difference for space derivative? If we have purely advective system: orCourant condition:

10. CE 498/698 and ERS 685 (Spring 2004) Lecture 1210 Weighted Differences i-1ii+1 whereand centered diff: forward diff: backward diff:

11. CE 498/698 and ERS 685 (Spring 2004) Lecture 1211 Solution stability Stability criterion: Numerical dispersion: spatialtemporal