A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.
In this paper, a Lagrangian of the coupled Navier-Stokes equations is
proposed based on the semi-inverse method. The fractional derivatives in the
sense of Riemann-Liouville definition are used to replace the classical
derivatives in the Lagrangian. Then the fractional Euler-Lagrange equation
can be derived with the help of the fractional variational principles. The
Agrawal?s method is devot?ed to lead to the time-space fractional coupled
Navier-Stokes equations from the above Euler-Lagrange equation. The solution
of the time-space fractional coupled Navier-Stokes equations is obtained by
means of RPS algorithm. The numerical results are presented by using exact
solutions.
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