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Numerical Methods for Water Quality Modeling Stability Criteria

Lecture 12: Control-Volume Approach in Water Quality Modeling, exploring finite difference approximations, stability criteria, and numerical dispersion. Concepts include FTCS method and Courant condition.

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Numerical Methods for Water Quality Modeling Stability Criteria

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

  2. t i,l+1 l+1 Dt i-1,l i,l i+1,l l Dx i,l-1 l-1 forward difference over time x i i-1 i+1 finite difference approximations Lecture 12

  3. t i,l+1 l+1 Dt i-1,l i,l i+1,l l Dx i,l-1 l-1 x i i-1 i+1 centered difference over space finite difference approximations Lecture 12

  4. t i,l+1 l+1 Dt i-1,l i,l i+1,l l Dx i,l-1 l-1 x i i-1 i+1 centered difference over space finite difference approximations Lecture 12

  5. t i,l+1 l+1 Dt i-1,l i,l i+1,l l Dx i,l-1 l-1 x i i-1 i+1 finite difference approximations backward difference over space Lecture 12

  6. forward difference over time centered difference over space centered difference over space finite difference approximations FTCS Lecture 12

  7. l+1 l i-1 i i+1 second derivative with space first derivative with time FTCS explicit method Lecture 12

  8. must be positive! Control-Volume Approach Lecture 12

  9. Assuming Q, E, V, k are constant: What if we used backward difference for space derivative? or Courant condition: Solution stability If we have purely advective system: Lecture 12

  10. where and and centered diff: forward diff: backward diff: Weighted Differences Lecture 12

  11. Stability criterion: spatial temporal Solution stability Numerical dispersion: Lecture 12

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