A 2D unstructured finite volume method (FVM) shallow water solver written in C++. Fluxes can be evaluated with the Lax–Friedrichs or the Roe method. Implicit and explicit time stepping is available. Expicit time stepping can be performed with the Euler, the Runge-Kutta 2nd-order, and the Runge-Kutta 4th-order methods. The jacobian used for the implicit time stepping is numerically evaluated and the linear system of equations is solved with the BiCGSTAB method. The Eigen C++ template library for linear algebra is used along with OpenMP.
Grid
Water height
Effect of the flux method; Gray surface is Lax-Friedrichs, red is Roe
Effect of the integration method; Gray surface is Euler, red is Runge-Kutta 2nd order, blue is Runge-Kutta 4th order
The code solves the following partial differential equations
where
Integrating the partial differential equations with respect to the finite volume V gives
which can be simplified to
,
Application of the divergence theorem gives
.
Further simplification leads to
The flux across the cell faces is approximated with the Roe or Lax-Friedrichs Riemann solvers. The conservative variables are assumed constant over the cell which results in a first-order spatial approximation. For higher-order approximation the gradients at each cell center must be calculated and used to reconstruct the values at the cell faces. To avoid spurious oscillations, due to discontinuities, a limiter must also be introduced.
The time derivative is approximated with the forward Euler method.
The resulting discretised PDE is then given by
An explicit scheme is obtained if the residual is evaluated at the current time level, n.
An implicit scheme is obtained if the residual is evaluated at the next time level, n+1. Since the residual at n+1 is unknown, Taylor series can be used to approximate it. The higher order terms in the Taylor series are neglected. The solution becomes more expensive as one needs to evaluate the derivative of the residual with respect to the vector of conserved variables. The current method uses finite differences to evaluate the Jacobian matrix.
The code requires the Eigen C++ template libary. Download the libary and extract its contents to a folder named Eigen inside the includes folder (includes/Eigen). Once extracted use make to compile the code. The sw_solver binary will be created in the bin folder.
To run the code navigate to the folder of the case e.g. dam_coarse_lf and run ../bin/sw_solver dam. The program will solve the dam break problem using the Lax-Friedrichs method for 1000 time steps with a time step of 0.003. The full list of options is:
- dt - timestep
- N_timesteps - number of timesteps
- riemann_solver - Riemman solver; 1 - Lax-Friedrichs, 2 - Roe
- output_frequency - Results output frequency; 0 - no output, i where i > 0 - every ith iteration
- integrator - Integration method; 0 - Euler, 1 - Runge-Kutta 2nd-order, 2 - Runge-Kutta 4th order
- implicit - Solve the equation explicitly or implicitly; 0 - Explicit, 1 - Implicit
Note: If the implicit option is set to 1 the integrator option is not used.