1. Shallow water equations in 1D: Method of characteristics
Numerical Hydraulics
Shallow water equations in 1D:
Method of characteristics
Wolfgang Kinzelbach with
Marc Wolf and
Cornel Beffa

2. Characteristic equations
As shown before, the St. Venant equations can be put into the form:
To obtain total differentials in the brackets we have to choose

3. Thus we obtain the characteristic equations:
along
along

4. Types of characteristics
Positive and negative characteristics for sub-critical, critical and super-critical flow: t P C+ C- E W x
Terminology: P, W (West) und E (East) instead of i, i-1, i+1

5. Integration of the characteristic equations
Multiplication with dt and integration
along characteristic line
and along characteristic line
yields:

6. Integration of the characteristic equationsor
This implies a linearisation. The wave
velocity becomes constant in the element.

7. Grid for subcritical flow (1)
Characteristics start on grid points
Zeit
Problem: Characteristics
intersect between grid points
in points P at time levels
which do not coincide with
the time levels of the grid.
Results have to be
interpolated. P
j+1 j W C E x
Terminology C (Center), W (West) und E (East) instead of i, i-1, i+1

8. Grid for subcritical flow (2)
Characteristic lines end at point P, starting points do not coincide with
grid points. Values at starting points are obtained by interpolation
from grid point values
Zeit
We choose this
variant! P
j+1 j C W E x
Terminology C (Center), W (West) und E (East) instead of i, i-1, i+1

9. Interpolation
Solve for two unknowns vL and cL

10. Starting point L
Solution for vL and cL yields:

11. Starting point R for subcritical flow
In analogy to point L, variables for point R vR and cR

12. Starting point R for supercritical flow
Starting point of characteristic between W and C
Using velocity v-c we obtain

13. Final explicit working equations
Integration from L to P and from R to P
with
2 equations with
2 unknowns
Boundary conditions
are required as discussed
in FD method.
As method is explicit
CLF-criterium applies.
and

14. Classical dam break problem: Solution with method of characteristics (Test problem)
Propagation velocity of fronts slightly too high

15. Matrix form of the St. Venant equations (1D)

16. Finite volume method FV formulation for this vector equation:
e and w designate the east and west boundary of the FV cell respectively w e
i-1 i
i+1 Dx

17. Finite volume method The computation of the term
can be done in different ways. E.g. with an upwind scheme (e becomes i and w becomes i-1 if the wave propagates in positive x-direction.)
For the time discretisation we choose an explicit method

18. Upwind formulation

19. Improvement of method
A further improvement can be reached by flux-limiting
The Roe-method is such an improvement. It can take into account discontinuities across the cell boundaries.

20. Flux difference splitting (Roe)
Idea: At the cell boundary the flux is computed according to the characteristics by a positive/negative linear wave (splitting).
The flux at the east side of a cell is: l r w e
i-1 i
i+1 or Dx

21. Flux difference splitting (Roe)
The Roe matrix A is the Jacobian matrix of the flux vector
The division into left and right part allows to account for discontinuities.

22. The Roe matrix
The Roe matrix can be computed as:

23. Fluxes according to Roe scheme
The fluxes at a cell side are computed from the left/right-side fluxes, e.g.:
and the variables on the new time level are:

24. Shallow water equations:
Advection:
Shallow water equations:
Fluxes:

25. Shallow water equations with first order upwind (Flux from left/west):