Revisiting well-posed boundary conditions for the shallow water equations

Abstract

We derive a general form of well-posed open boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. The boundary conditions are also discussed for a rectangular domain. We compare the results with a number of often used open boundary conditions and show that they are a subset of the derived general form.

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

“…If we can estimate the solution in all forms of data, the solution is strongly well-posed, if zero boundary data is necessary for obtaining an estimate, it is well posed see [43,44,45] for more details on well-posedness.…”

A new well-posed vorticity divergence formulation of the shallow water equations

“…The standard formulation (3) in terms of velocities and height of the shallow water equations is well established, and energy estimates as well as a general set of boundary conditions for the initial boundary value problem can be found, see for example (Ghader and Nordström (2014)). For that reason, we will in this paper focus on the formulation (7) and (12).…”

“…Oliger and Sundström (1978) derived well-posed boundary conditions for several sets of partial differential equations including the SWE by using the energy method. Ghader and Nordström (2014) derived a general form of well-posed open boundary conditions using similar techniques.…”

Comparative Study on the Open Boundary Conditions of Shallow Flows

“…Mathematical studies on OBC's well-posedness have usually been [e.g., 2,4,10,11,17]. Oliger and Sundström [10] formulated well-posed boundary conditions of the various PDEs using the energy method.…”

“…The energy method and maximally semi-bounded operators lead directly to well-posed problems [11]. In [17], Ghader and Nordström derived well-posed conditions for the linearized inviscid SWE and its OBC. They symmetrized the two-dimensional linearized inviscid SWE using similarity transformation and derived a well-posed boundary condition using the same method with [10] and definition 9.5.2 of [2] (See [17] for more details).…”