Two general avenues are available in order to obtain estimates for hyperbolic systems of conservation laws. They are the energy method of Kreiss [16, 17] and the entropy stability theory of Tadmor [32, 33]. Traditionally, the method of Kreiss has been applied to linearized versions of systems of hyperbolic equations in order to develop boundary treatments that lead to an energy estimate. In practice, these boundary conditions are needed to develop energy stable numerical approximations that weakly impose boundary information, e.g., through simultaneous approximation terms [4, 23, 24, 25] or numerical interface flux functions [14, 19, 40, 41]. In contrast, the method of Tadmor has been applied to nonlinear hyperbolic systems on domains with periodic boundary conditions (or infinite domains) in order to obtain entropy conservation. This makes the investigation of entropy conservation similar to the classical von Neumann stability analysis in the sense that boundaries are ignored [12, 29]. By adding dissipation, entropy stability is obtained for periodic or infinite domains . Hence, the use of entropy stability theory to develop provably stable boundary conditions has been limited [14, 27, 31]. The main strengths and weaknesses of these two approaches are: The energy analysis of Kreiss provides boundary conditions, but it is difficult to apply in the nonlinear case. It is straightforward to apply the entropy analysis of Tadmor for general systems of nonlinear equations, but it does not provide boundary condition information.
The focus of this work is to examine the connection between the classical energy method of Kreiss and Tadmor’s entropy analysis. Particular focus is given to comparing and contrasting these two strategies for deriving stable boundary treatments. In doing so, we demonstrate that the nonlinear energy method of Kreiss provides fundamental information about the hyperbolic system, that aid in choosing a minimal number of suitable boundary conditions, which is required for existence [11, 12, 25], and energy stability. The entropy stability theory of Tadmor is often portrayed as a nonlinear generalization of energy stability analysis . However, we show that it gives no information about the characteristics of the hyperbolic problem and offers little or even erroneous information as to what type of and how many boundary conditions are needed on a general bounded domain [30, 31]. We also include the linear energy analysis in our comparison and discuss its weaknesses.
To perform this comparison we consider the two-dimensional nonlinear shallow water equations (SWEs). Shallow water models are of particular interest for flow configurations where the vertical scales of motion are much smaller than the horizontal scales, such as in rivers or lakes [37, 38]
. The SWEs are a system of nonlinear hyperbolic partial differential equations that represent the conservation (or balance) of mass and momentum, depending on the forces, e.g. bottom friction,[6, 19, 37, 41]. An auxiliary conserved quantity, not explicitly built into the SWEs, is the total energy of the system. This additional conservation law can be used to create a stability estimate for the total energy [9, 34, 40] or build numerical approximations that respect the evolution of the total energy [8, 9, 22]. For the shallow water system the total energy also acts as a mathematical entropy function and fits into the entropy analysis framework of Tadmor .
Thus, the total energy and analysis of it for the SWEs act as a bridge between the classical energy method of Kreiss and Tadmor’s entropy stability theory. We apply the energy method [12, 16, 17] and derive a bound of the total energy, which for the SWEs is a particular scaled version of the norm of the solution. In the following, we will demonstrate that the energy method is consistent with the mathematical entropy analysis, but also that it provides additional information and guidance with respect to boundary treatments for the nonlinear problem. Investigations into energy consistent boundary conditions for the linearized SWEs are many [2, 3, 10, 20, 26, 37]. The linear analysis leads to a well-posed linear initial boundary value problem [10, 26, 37]. These linear boundary conditions can then be applied to the nonlinear case [10, 36]. However, as we will show, there are situations where the linear analysis cannot be applied to the nonlinear case. Similar to the entropy analysis, linear boundary treatments do not necessarily provide an energy estimate for the nonlinear problem.
The paper is organized as follows: The SWEs are given in Sect. 2. An estimate of the total energy for the shallow water system is provided in Sect. 3 using a minimal number of boundary conditions. In Sect. 4, we provide details, analysis, and discussion of the general open boundary conditions for the two-dimensional nonlinear SWEs in subcritical and supercritical flow regimes. In particular, we discuss the differences between the results from the linear energy analysis, the nonlinear energy analysis, and the entropy analysis. Concluding remarks are drawn in the final section.
2. Shallow water equations
We begin with the two-dimensional SWEs over a flat bottom topography written in conservative form 
which includes the continuity and momentum equations. Here is the water height, and are the fluid velocities in the - and -directions, and is the gravitational constant. The system of equations (2.1) are derived under the physical requirement that the water height [37, 38]. Additionally, we include the influence of Coriolis forces with the parameter which, for convenience, is assumed to be a constant. In practical applications is typically a function of latitude , which would not affect the subsequent energy analysis in this work.
In (2.2), we formulate the non-conservative equations in terms of the geopotential to simplify the analysis . Note, that according to the physical and mathematical requirements of the problem. Next, we write (2.2
) compactly in matrix-vector form by introducing the solution vectorand
The total energy (or entropy) of the SWEs is the sum of the kinetic and potential energy  where
that yield the total energy (or entropy) conservation law
In the entropy analysis of Tadmor, the total energy acts as a mathematical entropy function for the SWEs . As such, it is possible to define a new set of entropy variables where
). This contraction of the conservative form of the SWEs into entropy space involves the chain rule and relies on certain compatibility conditions between the conservative fluxes from (2.1) and the entropy fluxes (2.4) [8, 33].
3. Energy stability analysis
where is a constant independent of the solution . The matrix simultaneously symmetrizes the flux matrices and
To determine the scaling constant we examine the solution energy that will arise in the later analysis:
where . We want the solution energy to match the total energy (2.3)
Therefore, we take and the final symmetrization matrix reads
From the skew-symmetry of the Coriolis matrix we immediately see that. The flux matrices are now symmetrized and take the form
We seek to rewrite (3.2) with complete derivatives, and use the relations
The expression (3.2) becomes
We compute the derivatives of the matrices to be
which gives, noting the continuity equation from (2.2),
It is straightforward to compute
Therefore, (3.3) becomes
Remark 3.1 (Connection to mathematical entropy analysis).
It is interesting to examine how the statement (3.4) compares to the energy conservation law created with the analysis tools in . Due to the construction of the symmetrization matrix (3.1), we have the time evolution of the total energy
Next, we look at the energy flux contribution in the -direction
Similarly, in the -direction we find
Next, we examine the additional terms in (3.4) and try to incorporate them into the energy rate by rewriting the scalar terms into matrix-vector forms. We denote the required additional matrices by and , respectively. There are many possible ways to do this. We choose to follow a strategy where we require:
and to be simultaneously diagonalizable for any normal vector .
The scalar terms from (3.4) to be written in the normal direction as
The first requirement ensures that the matrices and
have the same eigenvectors. Combined with the matrix structure of the second requirement, this simplifies the derivation of boundary conditions, as will be demonstrated in Sect.4. To create the matrices and we need
Consider two real matrices . If either the matrix or has distinct eigenvalues and the matrices commute
then the matrices are simultaneously diagonalizable
See . ∎
In the current analysis we know that the eigenvalues of are all distinct. Therefore, due to Lemma 3.3, it is sufficient to guarantee simultaneous diagonalizability if we can determine a matrix that commutes with . This leads to the following result.
commutes with and is simultaneously diagonalizable with the same right eigenvector matrix , i.e.,
Additionally, the matrices
commute with and , respectively. Furthermore, we can reformulate the scalar terms in (3.4) to be
See Appendix A. ∎
Integrating (3.5) over , gives
We compactly write the time dependent term by introducing the norm
for the symmetric positive definite matrix . By applying Gauss’ theorem, we find
with the outward pointing normal vector on the surface . We rewrite the line integral contribution in the normal direction to be
with the matrix
and given in Theorem 3.4.
The eigenvalues of the matrices and are
where we define the wave celerity . From the construction of the matrix in Theorem 3.4 we know it has the same eigenvectors as , which are
This gives the eigendecomposition
From this information the expression (3.6) becomes
now written in terms of the scaled characteristic variables
where is the tangential velocity.
Remark 3.5 (Relation to linear analysis).
We can recover the characteristic variables from the work in  if we take
and then perform a linearization of the solution around a constant mean state.
The relation (3.7) implies that the nonlinear two-dimensional SWEs will be energy stable provided that the surface integral is made positive with a minimal number of energy stable boundary conditions.
Remark 3.6 (Entropy flux at the boundary).
A straightforward computation yields an alternative form of the boundary integral to be
This shows that the nonlinear energy analysis is consistent with a weak form of the entropy conservation law (2.5) . This form has been used to create entropy stable boundary conditions at non-penetrating walls [14, 27, 31]. However, as will be shown in Sect. 4.3, it offers no guidance as to what type of or how many boundary conditions should be imposed to guarantee energy or entropy stability, on general types of boundaries.
Remark 3.7 (Inconsistency with linear analysis).
If we use the linear analysis and neglect the terms contained in the matrices and , the contribution to the energy will contain an additional term at the boundary. In Sect. 4.3, we clarify what will happen if the linear results are applied in the nonlinear case.
4. Energy stable boundary conditions
The sign of the normal velocity determines whether there are inflow or outflow conditions at the domain boundary. That is, corresponds to inflow conditions and corresponds to outflow conditions. Additionally, we have that the argument of the surface integral from (3.7) is
where we introduce to be the new augmented eigenvalues.
Remark 4.1 (Solid wall boundary).
At a solid wall boundary (formally an outflow boundary) the normal velocity is zero and the statement (4.1) becomes
To determine energy stable open boundary conditions for a general domain
must hold. The boundary conditions must be of Dirichlet type. Neumann and Robin type boundary conditions are not admissible due to the lack of gradients. We rewrite condition (4.2) into the form
where contains the outgoing boundary information, the incoming information. The diagonal blocks and contain the positive and negative eigenvalues, respectively. Thi<s clear separation of the positive and negative eigenvalue contributions as well as the characteristic variables into incoming and outgoing boundary information provides a suitable setting to discuss energy stable boundary conditions for the nonlinear problem.
To start, we note that in order to bound (4.3), the minimal number of boundary conditions required is equal to the size of [23, 24]. The most general form of the boundary conditions is written [23, 24]
where is a coefficient matrix with the number of rows equal to the minimal number of required boundary conditions (equal to the size of ). The vector contains known external data from the boundary. Essentially, the boundary condition (4.4) represents the incoming information as a linear combination of the outgoing information and boundary data. With the rewritten stability condition (4.3) and the general boundary condition (4.4) we have
The general boundary conditions (4.4) for the nonlinear problem lead to energy stability and a bound provided
See . ∎
The proof in  for inhomogeneous boundary conditions involves some technical boundedness issues that we avoid in the current discussion. This is because it is straightforward to homogenize the boundary conditions [12, 29] and prove energy stability for the system augmented with a forcing function. Therefore, in the forthcoming analysis we take .
To present and discuss energy stable boundary conditions we introduce the Froude number for the normal flow component
which is a translation of the Mach number into the shallow water flow context [37, 41]. It serves to classify flow regimes as supercritical (or torrential) when and subcritical (or fluvial) when . Most shallow water flows are subcritical where, typically, . However, under special circumstances, like near a dam failure or over a non-constant bottom topography, the flow can become supercritical, e.g. [1, 5, 41].
The sign and magnitude of the normal flow velocity determine the number of boundary conditions that are needed. To construct energy stable boundary conditions we must satisfy (4.2) with a minimal number of boundary conditions, i.e. a minimal number of rows in [23, 24]. We collect the different cases:
- Supercritical inflow where :
The eigenvalues are all negative. This corresponds to
and we need three boundary conditions.
- Supercritical outflow where :
The three eigenvalues are positive such that
Therefore, zero boundary conditions are required.
- Subcritical inflow where :
We have and . Hence,
and we need two boundary conditions.
- Subcritical inflow where :
Here the eigenvalues are all negative, i.e., . The sign of changed due to the strength of the normal velocity component. We again have situation (4.6) and require three boundary conditions, as in the supercritical case.
- Subcritical outflow where :
We have and . Hence,
and we need one boundary condition.
- Subcritical outflow where :
All the eigenvalues are positive, i.e., . The sign of flipped due to the strength of the normal velocity component such that the characteristics match (4.7) and now zero boundary conditions are needed, as in the supercritical case.
Compared to the linear analysis, e.g. [3, 10, 37], we see that the relative magnitude of the normal velocity, , with regards to the wave celerity, , affect the number of boundary conditions differently. This is due to the augmentation of the eigenvalues in the stability condition (4.2) by the additional matrix terms given in Theorem 3.4. Interestingly, the number of boundary condition needed only changes in the subcritical flow regime. As an example, the boundary treatment of a subcritical flow match that of a supercritical flow for subcritical outflow where . As previously mentioned, most shallow water flows are subcritical, which highlights the importance of this new finding.
For non-penetrating walls, where , (4.10) is energy stable as discussed in Remark 4.1. This is formally a subcritical outflow boundary, so the number of boundary conditions, i.e. one, is correct. However, in the general case, the statement (4.11) gives, erroneously, that the number of boundary conditions is entirely defined by the sign of the normal velocity. It indicates that three boundary conditions are required for and zero boundary conditions for . We see that the wave celerity or the flow being classified as subcritical or supercritical do not even appear in the boundary treatment provided by the entropy analysis.
4.1. Supercritical inflow and outflow boundaries
The boundary treatment for supercritical flows for the nonlinear equations matches the linear analysis as well as the entropy analysis. For completeness, we restate the boundary conditions: At a supercritical inflow boundary we need three boundary conditions to satisfy (4.3). Because , the term from (4.4) with the coefficient matrix vanishes and the boundary condition has the form
or any such linear combination.
4.2. Subcritical inflow and outflow boundaries
First, we consider subcritical inflow where . The number of boundary conditions to specify depends on the magnitude of the Froude number.
If we are in the regime where , then we must prescribe two boundary conditions. Following the ansatz (4.4) the coefficient matrix is
such that the two incoming characteristic components, , are written in terms of the outgoing information, ,
with unknown coefficients and .
For a subcritical inflow with the sufficient condition for nonlinear energy stability from Theorem 4.2 becomes
At an inflow boundary, we know that the normal velocity is negative or . Thus, we can rewrite (4.15) into the form
We now know how to select energy stable values of and for inflow boundaries due to (4.16). Three special cases are:
, : This boundary condition sets the tangential velocity to zero and either prescribes the normal velocity component () or the geopotential (). These boundary conditions fail to satisfy the condition (4.14) since
However, the energy stability theory described herein only gives a sufficient condition for stability. Thus, these boundary conditions are not excluded out-right and, in fact, they are used in practice, e.g. [3, 35, 39]. Moreover, Oliger and Sundström , analyzed these boundary conditions using normal-mode analysis  and found that (specification of normal velocity) is stable, whereas (specification of water height) is not.
If we are in the regime where , then we need three boundary conditions also for the subcritical inflow. This corresponds to the supercritical inflow boundary conditions (4.12).
Next, we examine subcritical outflow where . First we consider flows with and must prescribe one boundary condition. Now there are two outgoing characteristic components, , and a single incoming one, . We take the coefficient matrix from (4.4) to be
and find the boundary term to have the form
For subcritical outflow with the sufficient condition for nonlinear energy stability from Theorem 4.2 becomes
Again, the constants and must be determined. The contribution (4.1) becomes
To guarantee energy stability, (4.19) must be positive for all values of and . This is true only if the expression is positive definite, i.e. the discriminant must be negative and the coefficients of the square terms positive. It turns out that the positivity of the square terms is included in the condition on the discriminant. Thus, it is sufficient to investigate
We expand and rearrange terms to find
Next, we divide by , substitute the form of the three propagation speeds, and rewrite the expression in terms of the Froude number (4.5)
Two important subsets of subcritical outflow boundary condition emerge:
: Here, the condition for energy stability becomes
This corresponds to a special choice of the boundary condition where either