Numerical Simulation Of A Runge-kutta Discontinuous Galerkin Scheme For One-dimensional Water Hammer Equations

The water hammer equations is in common use of the water hammer phenomena andis nonlinear partial differential equations of hyperbolic type. The classical method forsolving the water hammer equations: the characteristic line method, is not good at dealingwith complicated boundary conditions, so it is hard to solve the analytical solutions ofwater hammer equations which with the common conditions, and numerical value is animportant research on current form. In this paper we use the discontinuous finite elementsolving one dimension water hammer equations.The discontinuous finite element method is developed based on the finite elementmethod and finite volume method. It is good at dealing with the problem which with acomplex region boundary conditions or have the nature of problems with discontinuoussolutions, and it could parallel compute and have grid adaptive advantages. In the specificinitial conditions, the water hammer equations have discontinuous properties,so we use thediscontinuous finite element solving one dimension water hammer equations, so choosingthe discontinuous finite element method for solving the water hammer equation can makeup the shortcomings of the classical method, and it have direct benefits for solvinghigh-dimensional water hammer equation in the further.The main contents are as follows: First, a brief introduction to the discontinuous finiteelement method. Second, use it solving the water hammer equations. In the process havefour main factors affecting the accuracy of the solution: the numerical flux, slopeconstraint, discrete time format, boundary conditions. In this paper, there are detailedintroduction and gives each algorithm for water hammer equations of each fatcor. In thesolution process, correctly handle the boundary condition of water hammer equations is thekey to success,it is the main contribution of this paper. In the final, an example of twocases(continuous solution and discontinuous solution) compute numerical solution andcompare to the results computed by the characteristic line method. The experiment wascarried out by four influence factor of each case, and analysis of advantages and disadvantages, then to find the best way to solve the water hammer equations. After this works, we can draw the following conclusion: the discontinuous finiteelement method can deal with the water hammer equations very well, and the accuracy ofnumerical solutions can achieve the accuracy of the characteristic line method.