Abstract
This work presents anÂ enriched Galerkin (EG) discretization for the twodimensional shallowwater equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, elementwise constant functions. Similar to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallowwater equations is wellbalanced, in the sense that it preserves lakeatrest configurations. We evaluate the methodâ€™s robustness and accuracy using various analytical and realistic problems and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLABÂ /Â GNUÂ OctaveÂ framework FESTUNG.
1 Introduction
The two dimensional shallowwater equations (SWE) are used for aÂ wide range of applications in environmental and hydraulic engineering, oceanography, and many other areas. They are discretized on computational domains that can be very large and often feature complex geometries; therefore, the numerical schemes must be computationally efficient and robust. The nonlinearity and hyperbolic character of the SWE system constitute additional challenges for designing discretizations and solution algorithms, while other applicationspecific aspects such as local conservation of unknown quantities and wellbalancedness represent further desirable properties [seeÂ Hinkelmann etÂ al. (2015)] for aÂ brief overview of key requirements for SWE models).
The aforementioned issues led to a large number of studies dedicated to the development, analysis, and practical evaluation of various numerical techniques for solving the SWE. The earliest models used finite differences on structured grids, but, with the emergence of unstructuredmesh models [e.g.Â TELEMACÂ (Galland etÂ al. 1991) or ADCIRCÂ (Luettich etÂ al. 1992)], finite elements and finite volumes became the defacto standard. AÂ big advantage of the finite element approach is its potential to naturally accommodate higherorder discretizations on unstructured meshes; in this vein, various methods based on the continuous Galerkin (CG) and discontinuous Galerkin (DG) approximation spaces (or mixtures of both) have been described and compared in the literatureÂ (Hanert etÂ al. 2003; Comblen etÂ al. 2010). The results of these comparisons can be summarized (in aÂ somewhat oversimplified fashion) as follows:

Using CG for elevation and velocity is computationally very efficient (at least the loworder schemes) but tends to have stability issues usually represented by spurious elevation or velocity modes that arise from the LBB condition or from too large elevation spaces;

Using DG spaces for both unknowns is robust and needs no additional stabilization but may turn out to be computationally expensive [up to aÂ factor of four to five longer serial execution times compared to CG for piecewise linears on the same meshÂ (Dawson etÂ al. 2006)]. However, DG discretizations deliver higher accuracy than their CG counterpartsÂ (Kubatko etÂ al. 2009), are locally conservative, have better support for adaptive and nonconforming meshes, and can partially offset their computational costs by more efficient parallel scalingÂ (Kubatko etÂ al. 2009);

Some combinations of continuous and discontinuous spaces such as the lowestorder Raviartâ€“Thomas spaces are robust and computationally efficientÂ (Hanert etÂ al. 2003) but difficult to generalize to higher orders on triangular meshes.
The idea of the enriched Galerkin (EG) method is to enhance the CG approximation space using elementlocal discontinuous functions and, by relying on aÂ solution procedure nearly identical to that of the DG method (i.e. edge fluxes, Riemann solvers, etc.), produce aÂ robust, locally conservative discretization with fewer unknowns than aÂ DG discretization of the same order.
In its original form, the EG method adds aÂ piecewise constant DG component to aÂ continuous piecewise linear or multilinear CG space and uses the DG bilinear form. This method was introduced for the linear advectionâ€“diffusionâ€“reaction equation inÂ Becker etÂ al. (2003) and was proved there to be stable and to converge at the same rate as the piecewise linear DG discretization. The piecewise constant enrichment of the linear CG approximation makes the scheme intrinsically stable and imparts the local conservation propertyÂ (Sun and Liu 2009). Since then, the EG method has been generalized to CG spaces of the polynomial order k augmented by discontinuous, elementwise constant functionsÂ (Sun and Liu 2009; Lee etÂ al. 2016) and even to arbitrary enrichments with polynomials of degree m with \(1 \le m \le k\) (\(m=1\) indicates no enrichment at all) as discussed inÂ Rupp and Lee (2020). Arbitrary enrichments allow to consider EG as aÂ generalization of both CG and DG since EG uses the same bilinear and linear forms as DG.
While inheriting many advantages of DG, the EG method in multiple dimensions needs fewer degrees of freedom (DOF) than the DG method if \(m < k\). The standard EG method (with \(m = 0\)) has been developed to solve general elliptic and parabolic problems with dynamic mesh adaptivityÂ (Choi and Lee 2019; Lee etÂ al. 2018; Lee and Wheeler 2017, 2018) and was extended to address multiphase fluid flow problemsÂ (Lee and Wheeler 2018). Recently, the EG method has been applied to solve the nonlinear poroelastic problemÂ (Choo and Lee 2018; Kadeethum etÂ al. 2019), and its performance has been compared to other two and threefield methodsÂ (Kadeethum etÂ al. 2019). Another generalization of the EG approach considers enrichments by discontinuous polynomials defined on aÂ subcell meshÂ (Rupp etÂ al. 2020).
The EG methodology should not be confused with numerous other approximation space enrichment approaches such as bubble functions or eXtended Finite Element Method (XFEM) schemes. Main differences are the purpose (robustness and local conservation for EG vs. higher accuracy for bubbles and XFEM) and the underlying framework (DG vs. CG). Similarly to DG, the EG method also appears to be wellsuited for computational enhancements such as hybridization or static condensationâ€”this topic deserves aÂ separate study.
In the context of convectiondominated problems, additional stabilization techniques are often required. Classical stabilizations include the streamline upwind Petrovâ€“Galerkin (SUPG) method, weighted essentially non oscillatory (WENO) schemes, as well as provably boundpreserving alternatives [e.g.Â Brooks and Hughes (1982), Shu (2009), Cockburn and Shu (1998), Zhang and Shu (2011), Dumbser etÂ al. (2014)]. Popular methods designed particularly for DG discretizations include the edgebased Barthâ€“Jesperson limiterÂ (Barth and Jespersen 1989), and its vertexbased counterpartsÂ (Kuzmin 2010; Aizinger 2011). Limiters for an EG discretization of the linear advection equation have recently been proposed inÂ Kuzmin etÂ al. (2020). The approach therein deviates from classical DG slope limiters but rather fits in the framework of algebraic flux correctionÂ (Kuzmin 2012), which only recently has been extended to the DG settingÂ (Anderson etÂ al. 2017; Hajduk etÂ al. 2020). Numerical solutions based on the methods inÂ Kuzmin etÂ al. (2020) can be proven to satisfy discrete maximum principles under CFLlike time step restrictions, which makes the approach superior to geometrical slope limiting.
The main focus of the present work is to formulate and evaluate anÂ EG scheme for the SWE and to compare the quality of the EG and DG discretizations. The method is implemented in the FESTUNG frameworkÂ (Frank etÂ al. 2015; Reuter etÂ al. 2016; Jaust etÂ al. 2018; Reuter etÂ al. 2020) by modifying our DG implementation for the SWE presented inÂ Reuter etÂ al. (2019), Hajduk etÂ al. (2018). The same scheme was initially introduced in our UTBEST solverÂ (Dawson and Aizinger 2002; Aizinger and Dawson 2002) and later extended to three dimensions in UTBEST3DÂ (Dawson and Aizinger 2005; Aizinger etÂ al. 2013; Reuter etÂ al. 2015).
The paper is structured as follows. The mathematical model and its discretization using the EG method are presented in Sect.Â 2, aÂ brief description of the implementation using our MATLABÂ /Â GNUÂ OctaveÂ framework FESTUNG is the subject of Sect.Â 3. In Sect.Â 4, we demonstrate the accuracy and robustness of our EG scheme using anÂ analytical convergence test, aÂ supercritical flow example with discontinuous solution, and aÂ realistic tidal flow scenario for Bahamas islands. AÂ short conclusions and outlook section wraps up this work.
2 EG formulation for the SWE
2.1 Governing equations
The SWE in conservative form are given by
They are considered on a twodimensional, polygonallybounded domain \(\Omega \) and finite time interval \(\left( t_0,\,t_\mathrm {end}\right) \). By \(\xi \), we denote the free surface elevation of the water body with respect to a certain zero level (e.g., the mean sea level). The quantity \(H = \xi  z_\mathrm {b}\) represents the total fluid depth with \(z_\mathrm {b}\) denoting the bathymetry. \(\varvec{q} :=(U,V)^\mathsf {T}\) is the depth integrated horizontal velocity field, \(f_\mathrm {c}\) the Coriolis coefficient, g the gravitational acceleration, and \(\tau _{\text {bf}}\) the bottom friction coefficient. Wind stress, the atmospheric pressure gradient, and tidal potential are combined in the body force term \(\varvec{F} :=(F_x,F_y)^\mathsf {T}\).
Defining \(\varvec{c} :=(\xi , U,V)^\mathsf {T}\), systemÂ (1) can be rewritten in the following compact form:
with
and
In this work, we use several types of boundary conditions for the SWE (1). Following the standard approach, interior values are used as boundary values at parts of the boundary, on which the corresponding unknowns are not prescribed. ByÂ \(\hat{\cdot }\) , we denote prescribed boundary values of the respective unknowns. The following types of boundary conditions are used in this work:
Dirichlet boundary: Here, all unknowns are specified:
Land boundary: Denoting by \(\varvec{n}\) the exterior unit normal to \(\partial \Omega \), we set the normal flux to zero:
Open sea boundary: We prescribe the free surface elevation:
Radiation boundary: No unknowns are specified (free outflow). Finally, initial conditions are set for elevation and depth integrated velocity
2.2 Enriched Galerkin discretization
Let \(\{\mathcal {T}_\Delta \}_{\Delta >0}\) be a simplicial, shaperegular, quasi uniform, geometrically conformal family of triangulations of \(\Omega \subset \mathbb {R}^2\) with \(\# T\) denoting the total number of elements of \(\mathcal {T}_\Delta \). We obtain the local variational formulation of system (2) by multiplying with smooth test functions \(\varvec{\phi } \in C^\infty (\Omega )^3\) and integrating by parts on each element \(T \in \mathcal {T}_\Delta \) yielding
where we write \((\cdot ,\cdot )_{T}\) and \(\langle \cdot ,\cdot \rangle _{\partial T}\) for the \(L^2\)scalar products on elements and their boundaries, and denote by \(\varvec{n}=(n_x, n_y)^\mathsf {T}\) anÂ exterior unit normal to \(\partial T\).
Defining the broken polynomial spaces of order \(m \in \mathbb {N}_0\) as
and setting \(\mathbb P_{1}(\mathcal T_\Delta ) :=\{0\}\), we specify the EG test and trial spaces as
for integers \(1 \le m \le k\), \(k > 0\). Obviously, \(\mathbb P_m(\mathcal T_\Delta ) \subset \mathbb P_{k,m}(\mathcal T_\Delta ) \subset \mathbb P_k(\mathcal T_\Delta )\). Here, â€˜+â€™ denotes the sum of subspaces which is not aÂ direct sum if \(m \ne 1\). Examples of spaces are given in Fig.Â 1.
FromÂ (10) it follows immediately that
This dimension formula will come handy in Sect.Â 3 for defining EGÂ shape functions. More details about these spaces and their properties can be found inÂ Rupp and Lee (2020).
To obtain the semidiscrete EG formulation ofÂ (9), \(\varvec{c}\) and \(\varvec{\phi }\) are replaced by their discrete counterparts \(\varvec{c}_\Delta ,\varvec{\phi }_\Delta \in \mathbb P_{k,m}(\mathcal T_\Delta )^3\). Since the values of a discontinuous function are not unique on element edges, we replace the boundary term \(\varvec{A}(\varvec{c})\cdot \varvec{n}\) by aÂ numerical flux \(\varvec{\hat{A}}(\varvec{c}_\Delta ,\varvec{c}^+_\Delta , \varvec{n})\) that depends on the discontinuous values of the solution on element T (without superscript) and its edge neighbor (superscriptÂ \(^+\)). On domain boundaries, the specified boundary values of free surface elevation and velocity are utilized in place of \(\varvec{c}^+_\Delta \) for the flux computation. Finally, summing up over all elements \(T \in \mathcal {T}_\Delta \) yields
In this work, we use the Laxâ€“Friedrichs flux combined with the Roeâ€“Pike averagingÂ (Aizinger and Dawson 2002; Roe 1981; Roe and Pike 1984) defined as
where \(\lambda = \lambda (\varvec{c}_\Delta ,\varvec{c}_\Delta ^+,\varvec{n})\) is given by
with \(H_\Delta ^+ :=\xi _\Delta ^+  z_\mathrm {b}\) (note that the bathymetry \(z_\mathrm {b}\) is assumed to be continuous).
Discrete initial conditions are obtained using suitable projections of \(\xi _0\) and \(\varvec{q}_0\) into the discrete function spaceÂ (10) (cf.Â Rupp and Lee (2020), (Sect.Â 5) for aÂ comparison of reasonable choices for such projections). For the temporal discretization, we use strong stability preserving (SSP) explicit Rungeâ€“Kutta methodsÂ (Gottlieb etÂ al. 2001).
This study uses theÂ inviscid system of SWEÂ (1) (similarly to our DG scheme for the SWEÂ (Hajduk etÂ al. 2018), the EG formulation needs no viscous stabilization). Such terms can be easily added in the same way as in DG methods [see, e.g.Â Lee etÂ al. (2016)].
3 Implementation aspects
Our implementation of the EG method is based on the DG code for the SWE realized within the FESTUNG frameworkÂ (Reuter etÂ al. 2019; Frank and Reuter 2020). To obtain the required EG operators, we use aÂ simple strategy of modifying the DG code that exploits the fact that the EG (or, for that matter, also the CG) approximation spaces are embedded in the DG spaces of the same order. Therefore, all EG or CG basis functions can be represented as linear combinations of DG basis functions. This representation can be written in the form of aÂ (generally rectangular) system of linear equations, which can then be used to convert an available DG operator into the corresponding one for EG or CG spaces. This trick is by no means restricted to the SWE system; it can be just as easily applied to any linear or nonlinear PDE. The downside of this approach is that aÂ full DG discretization must be assembled firstâ€”thus limiting the efficiency gain due to aÂ smaller approximation space of the EG method. While simplifying switching between different DG, EG, and CG schemes, this approach is not recommended if aÂ dedicated EG scheme were to be implemented from scratch. However, our implementation uses for this purpose multiplication with aÂ timeindependent matrix (highly optimized operation in MATLABÂ /Â GNUÂ Octave) that only incurs aÂ negligible runtime overhead.
In this work, we utilize finite element spaces of polynomial degree \(k \le 2\). First consider the DG space \(\mathbb P_k(\mathcal T_\Delta )\) and denote by \(N:=\dim \mathbb P_k(\mathcal T_\Delta )\) the number DG unknowns for a scalar quantity. We have \(\dim \mathbb P_1(\mathcal T_\Delta ) = 3\,\#T\) and \(\dim \mathbb P_2(\mathcal T_\Delta ) = 6\,\#T\). For fixed k, let \(\left\{ \varphi _i: i\in \{1,\ldots ,N\}\right\} \) be the elementwise continuous nodal basis of the DG method, where the nodal property is local to every element \(T \in \mathcal T_\Delta \). Moreover, we denote by \(\left\{ \phi _i: i\in \{1,\ldots ,M\}\right\} \) with \(M:=\dim \mathbb P_{k,m}(\mathcal T_\Delta )\) a basis of the EG space.
In the following, we construct bases for various EG spaces from CG and DG basis functions. Here, one has to be careful since, in general, simply combining aÂ CG basis with elementwise DG basis functions produces aÂ linearlydependent set. For the admissible combinations of k and m with \(1\le m< k \le 2\) considered in our work, the EG bases can be constructed as follows:

\(k \in \{1,2\},~m=1\): The space \(\mathbb P_{k,1}(\mathcal {T}_\Delta )\) coincides with the CG ansatz space of order at most k. Therefore, we can simply choose the CG basis functions.

\(k=1,~m=0\): The union of characteristic functions of all elements \(T \in \mathcal T_\Delta \) and the continuous, piecewise linear interpolation functions for all but one mesh vertex form a basis.

\(k=2,~m=0\): We obtain a basis from the union of characteristic functions of all elements and all but one of the shape functions for the quadratic CG space.

\(k = 2,~m=1\): We use the standard linear DG basis and extend it by the nodal quadratic shape functions equal to 1 at one edge midpoint, but omit the ones corresponding to cell vertices. The functions are linearly independent and are contained in the space \(\mathbb P_{2,1}(\mathcal {T}_\Delta )\). As the number of basis functions equals \(\dim \mathbb P_{2,1}(\mathcal {T}_\Delta ) = 3\,\#T+\,\#E\), it must be a basis of \(\mathbb P_{2,1}(\mathcal {T}_\Delta )\). Here, \(\#T\) and \(\#E\) denote the number of triangles and edges of \(\mathcal {T}_\Delta \), respectively. The dimension formula (11) ensures that this, in fact, constitutes a basis of \(\mathbb P_{2,1}(\mathcal T_\Delta )\).
3.1 Assembling nonlinear EG operators from DG operators
Next, we show how to obtain an EG from the corresponding DG operator. To simplify the presentation, we formulate our approach for the scalar case noting that the generalization to vector fields is straightforward. DG discretizations of nonlinear PDEs such as the SWE feature nonlinear operators of the form
where \(b: \mathbb P_k(\mathcal T_\Delta ) \times \mathbb P_k(\mathcal T_\Delta ) \rightarrow \mathbb {R}\) is linear in the second argument. Our goal is to form the EG operator
by making use of (14). Since \(\mathbb P_{k,m}(\mathcal T_\Delta )\subset \mathbb P_{k}(\mathcal T_\Delta )\), it is possible to express the EG basis functions as linear combinations of the DG basis
Next, we insert (16) into (15), obtaining
for \(i \in \{1,\ldots ,M\}\). By defining \(\varvec{C} :=\left( C_{kl}\right) _{kl}\), \(k \in \{ 1,\ldots ,M\}\), \(l \in \{ 1,\ldots ,N\}\), we can thus write
Employing (17), we assemble the terms corresponding to (15) by modifying an existing DG discretization of the operator (14) and computing the matrixÂ \(\varvec{C}\). Due to the above choice of EG and DG basis functions, the matrixÂ \(\varvec{C}\) can be determined from simple geometric considerations during preprocessing.
3.2 Assembly of the EG mass matrix
We consider the operator \(b_\mathrm M\) defined by \(b_\mathrm M(u_{\Delta },v_{\Delta }) :=\left( u_{\Delta }\,,\,v_{\Delta }\right) _{\Omega }\). Since \(b_\mathrm M\) is a bilinear form, we can write its induced operator (mass matrix) as
with \(\left( B_\mathrm {M}^{\mathrm {DG}}\right) _{ij} = \int _{\Omega } \varphi _j \varphi _i\, dx\) for \(i,j \in \{1,..,N\}\). The corresponding EG mass matrix
can be obtained from (17), and, in operator form, is given by
That is, for operators induced by bilinear forms, the assembly can be preprocessed.
4 Numerical results
In this section, we investigate the performance of the EG method using artificial and realistic test problems for the SWE. The main goals of these numerical studies can be summarized as follows:

Verify the expected rate of convergence against a manufactured analytical solution;

evaluate the solution quality for realistic benchmarks;

compare the EG and DG methods in terms of accuracy, stability, and robustness.
We denote the results obtained for different combinations of k and m by the corresponding finite element spaces as illustrated in Fig.Â 1. To simplify notation, we omit their dependency on \(\mathcal {T}_\Delta \). Hence, we write \(\mathbb P_{k,m}\) instead of \(\mathbb P_{k,m}(\mathcal T_\Delta )\) and use the convention that \(\mathbb P_{k,k}\) is the DG space of polynomial degree k. As temporal discretization, we use SSP Rungeâ€“Kutta time stepping schemes described inÂ Aizinger etÂ al. (2000) with \(s = k+1\) stages. The time step size depends on the specific test problem.
4.1 Analytical convergence test
In our first numerical experiment, we approximate aÂ smooth solution of the SWE on the domain \(\Omega :=(0,1000) \times (0,1000)\). The coarsest mesh (corresponding to level one) consists of 16Â triangular elements and is shown in Fig.Â 2 (left). To investigate the convergence behavior, we consider aÂ total of five meshes obtained from the coarsest mesh by uniform refinement via edge bisection.
Setting \(\tau _{\mathrm {bf}}\) and \(f_\mathrm {c}\) to zero and prescribing the bathymetry by
we utilize the following analytical solution
Using the method of manufactured solution, we substitute (18) into (2)â€“(4) to obtain aÂ forcing function for the righthand side; in addition, Dirichlet boundary conditions for each unknown arising from (18) are imposed on all boundaries. We solve the SWE for the time interval (0,Â 1000) using the time step size \(\Delta t=1/4\) in all considered scenarios. This value of \(\Delta t\) is sufficiently small to make temporal discretization errors negligible compared to spatial approximation errors. Solution parameters are chosen as \(C_1=0.3,\, C_2=0.2,\, C_3=0.2\).
The projected initial surface elevation for \(\mathbb P_{1,0}\) on the coarsest mesh is shown in Fig.Â 2 (middle). FigureÂ 2 (right) displays the numerical solution for the surface elevation using the same approximation (\(k=1,~m=0\)) on the finest mesh at the final time.
For all considered CG, EG, and DG methods (\(1\le m\le k\le 2\)), we list the \(L^2(\Omega )\) discretization errors at the final time along with the corresponding convergence rates in TableÂ 1 and plot them in Fig.Â 3. The results indicate that we obtain at least second order convergence for all methods. Third order convergence can be observed for \(k=2\) and \(m\in \{1,2\}\), but not for the CG approximation \(k=2, m=1\). The results for \(k=2\) and \(m=0\) are slightly better than without the piecewise constant enrichment, but do not exhibit third order accuracy. In conclusion, the EG scheme converges almost exactly as well as the DG method if \(m=k1\), \(k \in \{1,2\}\), although the absolute errors tend to be somewhat larger than for their DG counterparts. This is to be expected because EG has fewer unknowns, and even the projected exact solution in general becomes less accurate if the number of unknowns is reduced.
In TableÂ 2, we list the total numbers of degrees of freedom for all configurations. Note that, the DOF for the EG method with elementwise constant enrichment \((k \in \{1,2\},~m=0)\) has approximately half as many DOF as the DG method of order k. EG with \(k=2,~m=1\) has ca. three quarters of the DOF for the quadratic DG method. In conclusion, the EG method performs similarly to DG for smooth solutions while having significantly fewer DOF.
One has to note here that the relation between the computational cost (even in serial execution) and the number of degrees of freedom is not aÂ simple one. In aÂ timeexplicit DG or EG scheme, main cost factors are element and edge integrals that are not significantly cheaper for the EG method than for the DG one. The situation is more complicated in timeimplicit and semiimplicit cases (not evaluated in our study): The EG system is smaller than the DG one and whether it is faster to solve depends on aÂ number of different aspects such as the condition number, sparsity structure, preconditioner, solver, etc.
The focus of the present study is anÂ evaluation of the accuracy, stability, and robustness of the EG method for the SWE; any conclusions about the computational performance of this scheme are clearly outside of the scopeâ€”in particular, since our implementation is based on MATLABÂ /Â GNUÂ OctaveÂ and rides on top of the DG assembly for the SWE solver. However, to give anÂ indication of the computational costs associated with anÂ EG scheme, we list in TableÂ 3 the cumulative (summed over all time steps) solution times for systems with the DG mass matrix (\(\mathbb {P}_{1,1}\)), the EG mass matrix (\(\mathbb {P}_{1,0}\)), and the EG mass matrix (\(\mathbb {P}_{1,0}\)) lumped according to the scheme proposed inÂ Becker etÂ al. (2003). We see there that naively using the direct MATLABÂ solver (backslash operator that calls the Cholesky method) on the EG mass matrix produces aÂ rather inefficient and poorly scaling implementation, whereas system solves with blockdiagonal DG and lumped EG matrices scale much better. Another interesting point to note is the fact that lumping the mass matrix leads to some degradation of the methodâ€™s accuracy and its convergence.
4.2 Supercritical flow in a constricted channel
In order to demonstrate the stability and robustness of the EG method, we solve the supercritical flow problem proposed inÂ Zienkiewicz and Ortiz (1995). The computational domain is a channel whose lateral boundary walls are constricted on both sides with an angle of five degrees (cf.Â Fig.Â 4). This benchmark uses constant bathymetry \(z_b \equiv 1\) while parameters \(\tau _{\mathrm {bf}},\) and \(f_\mathrm {c}\) are once more set to zero. The following initial and boundary conditions are prescribed for this problem: Initially, we set the surface elevation and momentum to \(\xi _0\equiv 0,\) \(\varvec{q}_0 = (1,0)^\mathsf {T}\), respectively. Land boundary conditions are imposed at the lateral (wall) boundaries, while at the inlet (left), free surface elevation as well as velocity are for all times specified to be identical to their corresponding initial values. Finally, at the outlet (right), radiation boundary conditions are used. Denoting by u and H the axial velocity and water depth at the inlet, respectively, the flow regime is made supercritical by choosing the inlet Froude number \(\mathrm {Fr}\)
achieved by setting the gravitational constant \(g=0.16~\frac{m}{s^2}\).
The solution to this problem converges to a steadystate, for which an analytical solution is availableÂ (Ippen 1951). FigureÂ 4 illustrates the computational domain along with the unstructured mesh used in all computations.
We run all simulations to aÂ steadystate using pseudo time stepping with aÂ time step of \(\Delta t = 1/10\) and present the results for all schemes in Fig.Â 5. The steadystate surface elevation shown in Fig.Â 5 (top left) is discontinuous and displays interactions of waves reflected from the channel constrictions. FigureÂ 5 (top right to bottom right) depicts the steadystate solutions for all considered EG and DG methods (\(0\le m\le k\le 2\)). Since no limiter was used, all approximations exhibit spurious oscillations close to the discontinuities. We expect the results to improve and the numerical solutions to become boundpreserving if a limiter, such as the one developed inÂ Kuzmin etÂ al. (2020), is utilized (at least) for the free surface elevation.
The specific type of enrichment appears to play aÂ major role for the stability of EG schemes. Thus the results for enrichments with \(m=k1,~k \in \{1,2\}\) are almost indistinguishable from the corresponding DG results. In particular, the characteristic wave features such as positions and magnitudes of discontinuities are in good agreement with the analytical solution. On the other hand, the EG solution for \(k=2,~m=0\) exhibits severe oscillations not only in the vicinity of the discontinuities but also in the remainder of the computational domain. This phenomenon indicates that aÂ piecewise constant enrichment of the quadratic CG space may not be sufficient to obtain anÂ intrinsically stable scheme. However, anÂ enrichment by piecewise linear discontinuous functions seems to remedy this issue, which confirms the previous findings in Sect.Â 4.1 andÂ (Rupp and Lee 2020). The results for this benchmark suggest that optimal EG schemes (\(m=k1\)) possess similar stability properties to their DG counterparts while offering potential advantages in computational efficiency.
4.3 Tidal flow at Bahamas Islands
The next benchmark considered in this work involves aÂ tidal flow scenario around the Bahamas Islands based on the configuration, parameter values, and mesh presented inÂ Westerink etÂ al. (1989) for the Bight of Abaco. The domain geometry, bathymetry as well as boundary types are depicted in Fig.Â 6 (left), and Fig.Â 6 (right) shows the unstructured mesh used in all simulations. Furthermore, four recording stations also shown in Fig.Â 6 (left) are placed at the following locations \((38\,667,49\,333)\), \((56\,098,9\,613)\), \((41\,263,29\,776)\), and \((59\,594,41\,149)\) (coordinates in meters) to monitor the temporal evolution of surface elevation and depth integrated velocity.
For bottom friction, we use the standard quadratic friction law \(\tau _{\text {bf}} = C_f  \varvec{q}  /H^2\) [see e.g.Â Vreugdenhil (1994)] with coefficient \(C_f = 0.009\). The constant Coriolis parameter is set to \(3.19 \times 10^{5}~\mathrm {s}^{1}\). The following tidal forcing is prescribed at the open sea boundaryÂ (Kolar etÂ al. 1994):
with time t in hours. In reallife ocean simulations, the initial conditions are often unknown or very difficult to obtain, therefore aÂ cold start initialization is performed: The flow domain is assumed to be at rest initially (\(\xi _0 \equiv 0\), \({\varvec{q}}_0 \equiv (0,0)^T\)). Then, starting immediately at initial time \(t=0\), the tidal forcingÂ (19) is imposed at the open sea boundary.
The simulations are run for aÂ total of 12Â days using constant time step \(\Delta t = 15\) seconds for EG and DG discretizations corresponding to \(0\le m \le k \le 2\). In Figs.Â 7,Â 8 andÂ 9, we compare the numerical solutions for all considered EG and DG methods at the four recording stations.
Surface elevation results in Fig.Â 7 demonstrate excellent agreement for all EG and DG methods. The curves lie nearly on top of each other and no differences can be visually detected at this resolution. The differences in the surface elevation between the EG and DG approximations at the recording stations are on the order of \(10^{4}\) meters.
In Figs.Â 8 andÂ 9, we plot both velocity components at the recording stations. For the station two (Figs.Â 8,Â 9 top right), one can observe slight differences between the approximations of orders one and two. Such behavior is fully consistent with the station comparisons performed inÂ Aizinger and Dawson (2002). The effect of approximation order has greater magnitude than the differences between the EG and DG discretizations. Small differences between EG and DG discretizations of the same order can be observed in the plots at some locations. The deviations at the recording stations are on the order of \(10^{3}\) to \(10^{4}~\mathrm {m\,s^{1}}\).
In addition to testing the accuracy and robustness of the EG method, we also use the Bahamas example to verify the wellbalancedness of the method. For this purpose, our simulation were run for the lakeatrest configuration with open sea boundary condition set to zero. Just as expected, no spurious circulation emerged for any of the EG or DG configurations.
5 Conclusions and future work
In this work, we present the first enriched Galerkin discretization for the system of 2D shallowwater equations, evaluated its performance in analytical and realistic test problems, and compared the numerical results to those obtained using our discontinuous Galerkin solver. The results of our studies demonstrate that EG schemes with enrichments using discontinuous spaces of order one less than the order of the continuous space display accuracy and robustness on aÂ par with the corresponding DG discretizations. Similarly to DG, the EG method guarantees local conservation of all primary unknowns while the total number of degrees of freedom is substantially lower than that for the DG space of the same order. This makes the enriched Galerkin method aÂ very attractive candidate for solving the shallowwater equations and other nonlinear hyperbolic PDE systems.
Several interesting avenues of future research concerning the development of anÂ EG solver for the SWE present themselves at this time. Thus one could try the quadraturefree methodology to improve the computational efficiency of the methodâ€”similarly to our recent work for the DG SWE solver inÂ FaghihNaini etÂ al. (2020). Formulating the EG method for the SWE as aÂ timeimplicit or aÂ semiimplicit schemeâ€”especially in aÂ combination with anÂ efficient linear solver such as the hierarchical scale separation method either directlyÂ (Aizinger etÂ al. 2015) or in aÂ hybridized settingÂ (SchĂ¼tz and Aizinger 2017) appears to be particularly attractive due to similarities with the DG method, on one hand, and aÂ substantially smaller system size, on the other. Also EG schemes hold promise (perhaps even more so than the DG ones) for bathymetry reconstruction using modified shallowwater equationsÂ (Hajduk etÂ al. 2020).
References
Aizinger, V.: A geometry independent slope limiter for the discontinuous Galerkin method. In: Krause, E., Shokin, Y., Resch, M., KrĂ¶ner, D., Shokina, N. (eds.) Computational Science and High Performance Computing IV, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 115, pp. 207â€“217. Springer, Berlin (2011). https://doi.org/10.1007/9783642177705_16
Aizinger, V., Dawson, C.: A discontinuous Galerkin method for twodimensional flow and transport in shallow water. Adv. Water Resour. 25(1), 67â€“84 (2002). https://doi.org/10.1016/S03091708(01)000197
Aizinger, V., Dawson, C., Cockburn, B., Castillo, P.: The local discontinuous Galerkin method for contaminant transport. Adv. Water Resour. 24(1), 73â€“87 (2000). https://doi.org/10.1016/S03091708(00)000221
Aizinger, V., Proft, J., Dawson, C., Pothina, D., Negusse, S.: A threedimensional discontinuous Galerkin model applied to the baroclinic simulation of Corpus Christi Bay. Ocean Dyn. 63(1), 89â€“113 (2013). https://doi.org/10.1007/s1023601205798
Aizinger, V., Kuzmin, D., Korous, L.: Scale separation in fast hierarchical solvers for discontinuous Galerkin methods. Appl. Math. Comput. 266, 838â€“849 (2015). https://doi.org/10.1016/j.amc.2015.05.047.
Anderson, R., Dobrev, V., Kolev, T., Kuzmin, D., Quezada de Luna, M., Rieben, R., Tomov, V.: Highorder local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation. J. Comput. Phys. 334, 102â€“124 (2017). https://doi.org/10.1016/j.jcp.2016.12.031
Barth, T., Jespersen, D.: The design and application of upwind schemes on unstructured meshes. In: Proceedings of the AIAA 27th Aerospace Sciences Meeting, Reno (1989)
Becker, R., Burman, E., Hansbo, P., Larson, M.G.: A reduced P1discontinuous Galerkin method. Chalmers Finite Element Center Preprint 2003â€“13 (2003)
Brooks, A.N., Hughes, T.J.: Streamline upwind/PetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations. Comput. Methods Appl. Mech. Eng. 32, 199â€“259 (1982). https://doi.org/10.1016/00457825(82)900718
Choi, W., Lee, S.: Optimal error estimate of elliptic problems with Dirac sources for discontinuous and enriched Galerkin methods. Appl. Numer. Math. 150, 76â€“104 (2019). https://doi.org/10.1016/j.apnum.2019.09.010
Choo, J., Lee, S.: Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Comput. Methods Appl. Mech. Eng. 341, 311â€“332 (2018). https://doi.org/10.1016/j.cma.2018.06.022
Cockburn, B., Shu, C.W.: The RungeKutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199â€“224 (1998). https://doi.org/10.1006/jcph.1998.5892
Comblen, R., Lambrechts, J., Remacle, J.F., Legat, V.: Practical evaluation of five partly discontinuous finite element pairs for the nonconservative shallow water equations. Int. J. Numer. Methods Fluids 63(6), 701â€“724 (2010). https://doi.org/10.1002/fld.2094
Dawson, C., Aizinger, V.: The local discontinuous Galerkin method for advectiondiffusion equations arising in groundwater and surface water applications. In: Chadam, J., Cunningham, A., Ewing, R.E., Ortoleva, P., Wheeler, M.F. (eds.) Resource Recovery, Confinement, and Remediation of Environmental Hazards, pp. 231â€“245. Springer, New York (2002). https://doi.org/10.1007/9781461300373_13
Dawson, C., Aizinger, V.: A discontinuous Galerkin method for threedimensional shallow water equations. J. Sci. Comput. 22(1â€“3), 245â€“267 (2005). https://doi.org/10.1007/s1091500441393
Dawson, C., Westerink, J., Feyen, J., Pothina, D.: Continuous, discontinuous and coupled discontinuouscontinuous Galerkin finite element methods for the shallow water equations. Int. J. Numer. Methods Fluids 52(1), 63â€“88 (2006). https://doi.org/10.1002/fld.1156
Dumbser, M., Zanotti, O., LoubĂ¨re, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47â€“75 (2014). https://doi.org/10.1016/j.jcp.2014.08.009
FaghihNaini, S., Kuckuk, S., Aizinger, V., Zint, D., Grosso, R., KĂ¶stler, H.: Quadraturefree discontinuous Galerkin method with code generation features for shallow water equations on automatically generated blockstructured meshes. Adv. Water Resour. 138, 103552 (2020). https://doi.org/10.1016/j.advwatres.2020.103552
Frank, F., Reuter, B.: FESTUNG: The Finite Element Simulation Toolbox for UNstructured Grids (2020). https://github.com/FESTUNG
Frank, F., Reuter, B., Aizinger, V., Knabner, P.: FESTUNG: a MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part I: diffusion operator. Comput. Math. Appl. 70(1), 11â€“46 (2015). https://doi.org/10.1016/j.camwa.2015.04.013
Galland, J.C., Goutal, N., Hervouet, J.M.: TELEMAC: a new numerical model for solving shallow water equations. Adv. Water Resour. 14(3), 138â€“148 (1991). https://doi.org/10.1016/03091708(91)90006A
Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stabilitypreserving highorder time discretization methods. SIAM Rev. 43, 89â€“112 (2001). https://doi.org/10.1137/S003614450036757X
Hajduk, H., Hodges, B.R., Aizinger, V., Reuter, B.: Locally filtered transport for computational efficiency in multicomponent advectionreaction models. Environ. Model. Softw. 102, 185â€“198 (2018). https://doi.org/10.1016/j.envsoft.2018.01.003
Hajduk, H., Kuzmin, D., Aizinger, V.: Bathymetry reconstruction using inverse shallow water models: finite element discretization and regularization. In: van Brummelen, H., Corsini, A., Perotto, S., Rozza, G. (eds.) Numerical Methods for Flows: FEF 2017 Selected Contributions, pp. 223â€“230. Springer, Cham (2020). https://doi.org/10.1007/9783030307059_20
Hajduk, H., Kuzmin, D., Kolev, T., Abgrall, R.: Matrixfree subcell residual distribution for Bernstein finite element discretizations of linear advection equations. Comput. Method. Appl. M. 359, 112658 (2020). https://doi.org/10.1016/j.cma.2019.112658
Hanert, E., Legat, V., Deleersnijder, E.: A comparison of three finite elements to solve the linear shallow water equations. Ocean Model. 5(1), 17â€“35 (2003). https://doi.org/10.1016/S14635003(02)000124
Hinkelmann, R., Liang, Q., Aizinger, V., Dawson, C.: Robust shallow water models. Environ. Earth Sci. 74(11), 7273â€“7274 (2015). https://doi.org/10.1007/s1266501547641. (editorial)
Ippen, A.: Highvelocity flow in open channels: a symposium: mechanics of supercritical flow. Trans. Am. Soc. Civ. Eng. 116(1), 268â€“295 (1951)
Jaust, A., Reuter, B., Aizinger, V., SchĂ¼tz, J., Knabner, P.: FESTUNG: a MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, part III: hybridized discontinuous Galerkin (HDG) formulation. Comput. Math. Appl. 75(12), 4505â€“4533 (2018). https://doi.org/10.1016/j.camwa.2018.03.045
Kadeethum, T., Nick, H.M., Lee, S., Richardson, C.N., Salimzadeh, S., Ballarin, F.: A novel enriched Galerkin method for modelling coupled flow and mechanical deformation in heterogeneous porous media. In: 53rd US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association, New York, NY, USA (2019). ARMA20190228
Kadeethum, T., Nick, H., Lee, S.: Comparison of twoand threefield formulation discretizations for flow and solid deformation in heterogeneous porous media. In: 20th Annual Conference of the International Association for Mathematical Geosciences (2019)
Kolar, R., Westerink, J., Cantekin, M., Blain, C.: Aspects of nonlinear simulations using shallowwater models based on the wave continuity equation. Comput. Fluids 23(3), 523â€“538 (1994). https://doi.org/10.1016/00457930(94)900175
Kubatko, E., Bunya, S., Dawson, C., Westerink, J., Mirabito, C.: A performance comparison of continuous and discontinuous finite element shallow water models. J. Sci. Comput. 40(1), 315â€“339 (2009). https://doi.org/10.1007/s1091500992682
Kuzmin, D.: A vertexbased hierarchical slope limiter for adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233(12), 3077â€“3085 (2010). https://doi.org/10.1016/j.cam.2009.05.028. Finite Element Methods in Engineering and Science (FEMTEC 2009)
Kuzmin, D.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, R.L.D., Turek, S. (eds.) FluxCorrected Transport: Principles, Algorithms, and Applications, vol. 2, pp. 145â€“192. Springer, New York (2012)
Kuzmin, D., Hajduk, H., Rupp, A.: Locally boundpreserving enriched Galerkin methods for the linear advection equation. Comput. Fluids 205(104525), 15 (2020). https://doi.org/10.1016/j.compfluid.2020.104525
Lee, S., Wheeler, M.F.: Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization. J. Comput. Phys. 331, 19â€“37 (2017). https://doi.org/10.1016/j.jcp.2016.10.072
Lee, S., Wheeler, M.F.: Enriched Galerkin methods for twophase flow in porous media with capillary pressure. J. Comput. Phys. 367, 65â€“86 (2018). https://doi.org/10.1016/j.jcp.2018.03.031
Lee, S., Lee, Y.J., Wheeler, M.F.: A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems. SIAM J. Sci. Comput. 38(3), A1404â€“A1429 (2016). https://doi.org/10.1137/15M1041109
Lee, S., Mikelic, A., Wheeler, M.F., Wick, T.: Phasefield modeling of two phase fluid filled fractures in a poroelastic medium. Multiscale Model. Simul. 16(4), 1542â€“1580 (2018). https://doi.org/10.1137/17M1145239
Luettich, R., Westerink, J., Scheffner, N.: ADCIRC: an advanced threedimensional circulation model for shelves, coasts and estuaries, ReportÂ 1: theory and methodology of ADCIRC2DDI and ADCIRC3DL. Technical Report Dredging Research Program Technical Report DRP926, US Army Engineers Waterways Experiment Station, Vicksburg, MS (1992)
Reuter, B., Hajduk, H., Rupp, A., Frank, F., Aizinger, V., Knabner, P.: FESTUNG 1.0: overview, usage, and example applications of the MATLAB/GNU Octave toolbox for discontinuous Galerkin methods. Comput. Math. Appl. (2020). https://doi.org/10.1016/j.camwa.2020.08.018
Reuter, B., Aizinger, V., KĂ¶stler, H.: A multiplatform scaling study for an OpenMP parallelization of a discontinuous Galerkin ocean model. Comput. Fluids 117, 325â€“335 (2015). https://doi.org/10.1016/j.compfluid.2015.05.020
Reuter, B., Aizinger, V., Wieland, M., Frank, F., Knabner, P.: FESTUNG: a MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part II: advection operator and slope limiting. Comput. Math. Appl. 72(7), 1896â€“1925 (2016). https://doi.org/10.1016/j.camwa.2016.08.006
Reuter, B., Rupp, A., Aizinger, V., Frank, F., Knabner, P.: FESTUNG: a MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part IV: generic problem framework and modelcoupling interface. Commun. Comput. Phys. 28, 827â€“876 (2020). https://doi.org/10.4208/cicp.OA20190132
Roe, P., Pike, J.: Efficient construction and utilisation of approximate Riemann solutions. In: R. Glowinski, J.L. Lions (Eds.) Computing Methods in Applied Sciences and Engineering, pp. 499â€“518 (1984)
Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357â€“372 (1981)
Rupp, A., Hauck, M., Aizinger, V.: A subcellenriched Galerkin method for advection problems. Submitted (2020). arXiv:2006.09041
Rupp, A., Lee, S.: Continuous Galerkin and enriched Galerkin methods with arbitrary order discontinuous trial functions for the elliptic and parabolic problems with jump conditions. J. Sci. Comput. 84(9), 25 (2020). https://doi.org/10.1007/s10915020012554
SchĂ¼tz, J., Aizinger, V.: A hierarchical scale separation approach for the hybridized discontinuous Galerkin method. J. Comput. Appl. Math. 317, 500â€“509 (2017). https://doi.org/10.1016/j.cam.2016.12.018.
Shu, C.W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82â€“126 (2009). https://doi.org/10.1137/070679065
Sun, S., Liu, J.: A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method. SIAM J. Sci. Comput. 31(4), 2528â€“2548 (2009). https://doi.org/10.1137/080722953
Vreugdenhil, C.B.: Numerical Methods for ShallowWater Flow. Springer, New York (1994). https://doi.org/10.1007/9789401583541
Westerink, J.J., Stolzenbach, K.D., Connor, J.J.: General spectral computations of the nonlinear shallow water tidal interactions within the bight of abaco. J. Phys. Oceanogr. 19(9), 1348â€“1371 (1989). https://doi.org/10.1175/15200485(1989)019<1348:GSCOTN>2.0.CO;2
Zhang, X., Shu, C.W.: Maximumprinciplesatisfying and positivitypreserving highorder schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752â€“2776 (2011). https://doi.org/10.1098/rspa.2011.0153
Zienkiewicz, O.C., Ortiz, P.: A splitcharacteristic based finite element model for the shallow water equations. Int. J. Numer. Methods Fluids 20(8â€“9), 1061â€“1080 (1995). https://doi.org/10.1002/fld.1650200823
Acknowledgements
This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanyâ€™s Excellence Strategy EXC 2181/1  390900948 (the Heidelberg STRUCTURES Excellence Cluster).
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articleâ€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleâ€™s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Hauck, M., Aizinger, V., Frank, F. et al. Enriched Galerkin method for the shallowwater equations. Int J Geomath 11, 31 (2020). https://doi.org/10.1007/s13137020001677
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13137020001677
Keywords
 Enriched Galerkin
 Finite elements
 Shallowwater equations
 Discontinuous Galerkin
 Local conservation
 Ocean modeling
Mathematics Subject Classification
 65
 76
 86