Courant number criteria

In a backwards-in-distance solution for advective transport in the absence of dispersion or diffusion, the Courant criterion limits the time step. In one dimension, the grid Courant number is the number of nodal blocks the fluid traverses over a time step. By the Courant criterion, the Courant number Co must not exceed one, or... 

Numerical dispersion can be minimized in several ways. The nodal block spacing Ax can be set small by dividing the domain into as many blocks as practical. The value specified for Dl can be reduced to account for the anticipated numerical dispersion. And a time step can be chosen to give a grid Courant number Co as close to one as allowed by the stability criterion (Eqn. 20.32). 

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

According to this criterion for decreasing the distance step in half it is necessary to lower the time step by the factor of 4. In other cases as the criterion was used Courant number Co describing the number of cells, which water passes in one time step. This number should not exceed 1. Which meant that the algorithm is stable only when the time step is selected so that the liquid moves in time step no further than by the length of one nodal block. Usually used Co values are between 0.5 and 1. In the absence of flow and diffusion this criterion is determined by inequality... 

The stability analysis of the dispersion-correction numerical scheme shows that the stability criterion is < 0.67 and —0.125 < 7 < 0.083, where C/ is the Courant number. The lower limit of 7 imposes a limitation on the ratio of grid size to water depth as 1.27. This means that the uniform grid size greater than approximately 1.27 times of a local water depth must be employed for the stability of the present dispersion-correction scheme. To satisfy the stability criterion of the dispersion-correction finite-difference model, the range of applicable water depth is limited if a uniform grid size is used for varying water depth. For practical purposes, this stability criterion can be solved by imposing intentionally a limitation on the... 

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