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

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

Thus we obtain the characteristic equations: along along

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

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

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

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

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

Interpolation Solve for two unknowns vL and cL

Starting point L Solution for vL and cL yields:

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

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

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

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

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

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

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

Upwind formulation

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.

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

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.

The Roe matrix The Roe matrix can be computed as:

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:

Shallow water equations: Advection: Shallow water equations: Fluxes:

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