Numerical shockwave anomalies in the resolution of the Shallow Water Equations with bed variations

The presence of numerical shockwave anomalies appearing in the resolution of hyperbolic systems of conservation laws is a well-known problem in the scientific community. The most common anomalies are the carbuncle and the slowly-moving shock anomaly. They have been studied for decades in the framework of Euler equations, but only a few authors have considered such problems for the Shallow Water Equations (SWE). In this work, the SWE are considered and the aforementioned anomalies are studied. They arise in presence of hydraulic jumps, which are transcritical shockwaves mathematically modelled as a pure discontinuity. When solving numerically such discontinuities, an unphysical intermediate state appears and gives rise to a spurious spike in the momentum. This is observed in the numerical solution as a spike in the discharge appearing in the cell containing the jump. The presence of the spurious spike in the discharge has been taken for granted by the scientific community and has even become a feature of the solution. Even though it does not disturb the rest of the solution in steady cases, it produces an undesirable shedding of spurious oscillations downstream when considering transient events. We show how it is possible to define a coherent spike reduction technique that reduces the magnitude of this anomaly and ensures convergence to the exact solution with mesh refinement. Concerning the carbuncle, which may also appear in presence of strong hydraulic jumps, a combination of Riemann solvers is proposed to circumvent it. Also, it will be shown how there is still room from improvement when treating anomalies in moving hydraulic jumps over variable topography

Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography

When designing a numerical scheme for the resolution of conservation laws, the selection of a particular source term discretization (STD) may seem irrelevant whenever it ensures convergence with mesh refinement, but it has a decisive impact on the solution. In the framework of the Shallow Water Equations (SWE), well-balanced STD based on quiescent equilibrium are unable to converge to physically based solutions, which can be constructed considering energy arguments. Energy based discretizations can be designed assuming dissipation or conservation, but in any case, the STD procedure required should not be merely based on ad hoc approximations. The STD proposed in this work is derived from the Generalized Hugoniot Locus obtained from the Generalized Rankine Hugoniot conditions and the Integral Curve across the contact wave associated to the bed step. In any case, the STD must allow energy-dissipative solutions: steady and unsteady hydraulic jumps, for which some numerical anomalies have been documented in the literature. These anomalies are the incorrect positioning of steady jumps and the presence of a spurious spike of discharge inside the cell containing the jump. The former issue can be addressed by proposing a modification of the energy-conservative STD that ensures a correct dissipation rate across the hydraulic jump, whereas the latter is of greater complexity and cannot be fixed by simply choosing a suitable STD, as there are more variables involved. The problem concerning the spike of discharge is a well-known problem in the scientific community, also known as slowly-moving shock anomaly, it is produced by a nonlinearity of the Hugoniot locus connecting the states at both sides of the jump. However, it seems that this issue is more a feature than a problem when considering steady solutions of the SWE containing hydraulic jumps. The presence of the spurious spike in the discharge has been taken for granted and has become a feature of the solution. Even though it does not disturb the rest of the solution in steady cases, when considering transient cases it produces a very undesirable shedding of spurious oscillations downstream that should be circumvented. Based on spike-reducing techniques (originally designed for homogeneous Euler equations) that propose the construction of interpolated fluxes in the untrustworthy regions, we design a novel Roe-type scheme for the SWE with discontinuous topography that reduces the presence of the aforementioned spurious spike. The resulting spike-reducing method in combination with the proposed STD ensures an accurate positioning of steady jumps, provides convergence with mesh refinement, which was not possible for previous methods that cannot avoid the spike

Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in Navas-Montilla and Murillo (2015) 1]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases are constructed

A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations

Powerful numerical methods have to consider the presence of source terms of different nature, that intensely compete among them and may lead to strong spatiotemporal variations in the flow. When applied to shallow flows, numerical preservation of quiescent equilibrium, also known as the well-balanced property, is still nowadays the keystone for the formulation of novel numerical schemes. But this condition turns completely insufficient when applied to problems of practical interest. Energy balanced methods (E-schemes) can overcome all type of situations in shallow flows, not only under arbitrary geometries, but also with independence of the rheological shear stress model selected. They must be able to handle correctly transient problems including modeling of starting and stopping flow conditions in debris flow and other flows with a non-Newtonian rheological behavior. The numerical solver presented here satisfies these properties and is based on an approximate solution defined in a previous work. Given the relevant capabilities of this weak solution, it is fully theoretically derived here for a general set of equations. This useful step allows providing for the first time an E-scheme, where the set of source terms is fully exercised under any flow condition involving high slopes and arbitrary shear stress. With the proposed solver, a Roe type first order scheme in time and space, positivity conditions are explored under a general framework and numerical simulations can be accurately performed recovering an appropriate selection of the time step, allowed by a detailed analysis of the approximate solver. The use of case-dependent threshold values is unnecessary and exact mass conservation is preserved

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb C}^2$ is exactly $\sqrt[4]{\frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${\mathbb R}^2$ of higher degrees.Comment: 16 page