246 research outputs found
A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids
A novel wetting and drying treatment for second-order Runge-Kutta
discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water
equations is proposed. It is developed for general conforming two-dimensional
triangular meshes and utilizes a slope limiting strategy to accurately model
inundation. The method features a non-destructive limiter, which concurrently
meets the requirements for linear stability and wetting and drying. It further
combines existing approaches for positivity preservation and well-balancing
with an innovative velocity-based limiting of the momentum. This limiting
controls spurious velocities in the vicinity of the wet/dry interface. It leads
to a computationally stable and robust scheme -- even on unstructured grids --
and allows for large time steps in combination with explicit time integrators.
The scheme comprises only one free parameter, to which it is not sensitive in
terms of stability. A number of numerical test cases, ranging from analytical
tests to near-realistic laboratory benchmarks, demonstrate the performance of
the method for inundation applications. In particular, super-linear
convergence, mass-conservation, well-balancedness, and stability are verified
Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Schemes for the Euler Equations under Gravitational Fields
This paper designs and analyzes positivity-preserving well-balanced (WB)
central discontinuous Galerkin (CDG) schemes for the Euler equations with
gravity. A distinctive feature of these schemes is that they not only are WB
for a general known stationary hydrostatic solution, but also can preserve the
positivity of the fluid density and pressure. The standard CDG method does not
possess this feature, while directly applying some existing WB techniques to
the CDG framework may not accommodate the positivity and keep other important
properties at the same time. In order to obtain the WB and
positivity-preserving properties simultaneously while also maintaining the
conservativeness and stability of the schemes, a novel spatial discretization
is devised in the CDG framework based on suitable modifications to the
numerical dissipation term and the source term approximation. The modifications
are based on a crucial projection operator for the stationary hydrostatic
solution, which is proposed for the first time in this work. This novel
projection has the same order of accuracy as the standard -projection, can
be explicitly calculated, and is easy to implement without solving any
optimization problems. More importantly, it ensures that the projected
stationary solution has the same cell averages on both the primal and dual
meshes, which is a key to achieve the desired properties of our schemes. Based
on some convex decomposition techniques, rigorous positivity-preserving
analyses for the resulting WB CDG schemes are carried out. Several one- and
two-dimensional numerical examples are performed to illustrate the desired
properties of these schemes, including the high-order accuracy, the WB
property, the robustness for simulations involving the low pressure or density,
high resolution for the discontinuous solutions and the small perturbations
around the equilibrium state.Comment: 57 page
Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography
A well-designed numerical method for the shallow water equations (SWE) should
ensure well-balancedness, nonnegativity of water heights, and entropy
stability. For a continuous finite element discretization of a nonlinear
hyperbolic system without source terms, positivity preservation and entropy
stability can be enforced using the framework of algebraic flux correction
(AFC). In this work, we develop a well-balanced AFC scheme for the SWE system
including a topography source term. Our method preserves the lake at rest
equilibrium up to machine precision. The low-order version represents a
generalization of existing finite volume approaches to the finite element
setting. The high-order extension is equipped with a property-preserving flux
limiter. Nonnegativity of water heights is guaranteed under a standard CFL
condition. Moreover, the flux-corrected space discretization satisfies a
semi-discrete entropy inequality. New algorithms are proposed for realistic
simulation of wetting and drying processes. Numerical examples for well-known
benchmarks are presented to evaluate the performance of the scheme
Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations
In this paper, we propose a well-balanced fifth-order finite difference
Hermite WENO (HWENO) scheme for the shallow water equations with non-flat
bottom topography in pre-balanced form. For achieving the well-balance
property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput.
Phys., 208 (2005), 206-227.] to balance the flux gradients and the source
terms. The fluxes in the original equation are reconstructed by the nonlinear
HWENO reconstructions while other fluxes in the derivative equations are
approximated by the high-degree polynomials directly. And an HWENO limiter is
applied for the derivatives of equilibrium variables in time discretization
step to control spurious oscillations which maintains the well-balance
property. Instead of using a five-point stencil in the same fifth-order WENO-XS
scheme, the proposed HWENO scheme only needs a compact three-point stencil in
the reconstruction. Various benchmark examples in one and two dimensions are
presented to show the HWENO scheme is fifth-order accuracy, preserves
steady-state solution, has better resolution, is more accurate and efficient,
and is essentially non-oscillatory.Comment: 24 pages, 11 figure
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms
Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations
The shallow water flow model is widely used to describe water flows in
rivers, lakes, and coastal areas. Accounting for uncertainty in the
corresponding transport-dominated nonlinear PDE models presents theoretical and
numerical challenges that motivate the central advances of this paper. Starting
with a spatially one-dimensional hyperbolicity-preserving,
positivity-preserving stochastic Galerkin formulation of the
parametric/uncertain shallow water equations, we derive an entropy-entropy flux
pair for the system. We exploit this entropy-entropy flux pair to construct
structure-preserving second-order energy conservative, and first- and
second-order energy stable finite volume schemes for the stochastic Galerkin
shallow water system. The performance of the methods is illustrated on several
numerical experiments
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