On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

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.

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A REDUCED STABILITY CONDITION FOR NONLINEAR RELAXATION TO CONSERVATION LAWS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891604000020 ◽

2004 ◽

Vol 01 (01) ◽

pp. 149-170 ◽

Cited By ~ 20

Author(s):

FRANÇOIS BOUCHUT

Keyword(s):

Conservation Laws ◽

Stability Condition ◽

Hyperbolic Systems ◽

Strong Stability ◽

Riemann Solvers ◽

Systems Of Conservation Laws ◽

Leading Term ◽

Relaxation Systems ◽

The One ◽

Approximate Riemann Solvers

We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore and Liu, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the Chapman–Enskog expansion. This reduced stability condition has the advantage of involving only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition. Our condition generalizes the one introduced by the author in the case of kinetic, i.e. diagonal semilinear relaxation. We provide an adapted stability analysis in the context of approximate Riemann solvers obtained via relaxation systems.

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