13 research outputs found
Well-balanced finite difference WENO schemes for the blood flow model
The blood flow model maintains the steady state solutions, in which the flux
gradients are non-zero but exactly balanced by the source term. In this paper,
we design high order finite difference weighted non-oscillatory (WENO) schemes
to this model with such well-balanced property and at the same time keeping
genuine high order accuracy. Rigorous theoretical analysis as well as extensive
numerical results all indicate that the resulting schemes verify high order
accuracy, maintain the well-balanced property, and keep good resolution for
smooth and discontinuous solutions
A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations
This work focuses on the numerical approximation of the Shallow Water
Equations (SWE) using a Lagrange-Projection type approach. We propose to extend
to this context recent implicit-explicit schemes developed in the framework of
compressibleflows, with or without stiff source terms. These methods enable the
use of time steps that are no longer constrained by the sound velocity thanks
to an implicit treatment of the acoustic waves, and maintain accuracy in the
subsonic regime thanks to an explicit treatment of the material waves. In the
present setting, a particular attention will be also given to the
discretization of the non-conservative terms in SWE and more specifically to
the well-known well-balanced property. We prove that the proposed numerical
strategy enjoys important non linear stability properties and we illustrate its
behaviour past several relevant test cases
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations
We provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography. A challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well-balancing of the approximate solution. A key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process.The underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. Furthermore we use the multiresolution analysis of the underlying data as an additional indicator of whether the limiter has to be applied on a cell or not. By this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution.By means of well-known 1D and 2D benchmark problems, we verify that multiwavelet-based grid adaptation can significantly reduce the computational cost by sparsening the computational grids, while retaining accuracy and keeping well-balancing and positivity