Numerical approximations of non-linear shallow water equations
Elfadil, Nosiba (2020)
Diplomityö
2020
School of Engineering Science, Laskennallinen tekniikka
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Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi-fe2020052739446
https://urn.fi/URN:NBN:fi-fe2020052739446
Tiivistelmä
In this work, the Finite Difference Methods (FDMs), and Method of Lines (MOL) have been used to solve the Shallow Water Equations (SWEs) in one dimension. The SWEs has a free surface with no rotation, it's kinematic viscosity is zero, it's pressure distribution is approximately hydrostatic, and it has a flat bottom topography with zero height.
The SWEs were derived by using the equations of the conservation of mass and momentum to have a non-linear partial differential equations (PDEs). The PDE problem was tackled by applying a number of numerical models. To solve it numerically, the Backward Euler (BE), Forward Euler (FE), Crank-Nicolson (CN), and Method of Lines (MOL) approximations were used for the time and space discretizations. Once the set of non-linear PDEs has been converted to a system of algebraic equations using different schemes, the algebraic equations were solved with the help of MATLAB, and the numerical results were compared with the analytic solution. The analytical solution was obtained using the eigenvalues and eigenvectors for the quasi-linear form of the non-linear PDEs.
The SWEs were derived by using the equations of the conservation of mass and momentum to have a non-linear partial differential equations (PDEs). The PDE problem was tackled by applying a number of numerical models. To solve it numerically, the Backward Euler (BE), Forward Euler (FE), Crank-Nicolson (CN), and Method of Lines (MOL) approximations were used for the time and space discretizations. Once the set of non-linear PDEs has been converted to a system of algebraic equations using different schemes, the algebraic equations were solved with the help of MATLAB, and the numerical results were compared with the analytic solution. The analytical solution was obtained using the eigenvalues and eigenvectors for the quasi-linear form of the non-linear PDEs.