138 research outputs found
On a numerical artifact of solving shallow water equations with a discontinuous bottom: Analysis and a nontransonic fix
In this paper, we study a numerical artifact of solving the nonlinear shallow
water equations with a discontinuous bottom topography. For various first-order
schemes, the numerical solution of the momentum will form a spurious spike at
the discontinuous points of the bottom, which should not exist in the exact
solution. The height of the spike cannot be reduced even after the mesh is
refined. For subsonic problems, this numerical artifact may cause the wrong
convergence to a function far away from the exact solution. To explain the
formation of the spurious spike, we perform a convergence analysis by proving a
Lax--Wendroff type theorem. It is shown that the spurious spike is caused by
the numerical viscosity in the computation of the water height at the
discontinuous bottom. The height of the spike is proportional to the magnitude
of the viscosity constant in the Lax--Friedrichs flux. Motivated by this
conclusion, we propose a modified scheme by adopting the central flux at the
bottom discontinuity in the equation of mass conservation, and show that this
numerical artifact can be removed in many cases. For various numerical tests
with nontransonic Riemann solutions, we observe that the modified scheme is
able to retrieve the correct convergence.Comment: 37 page
Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering
In highly diffusion regimes when the mean free path tends to
zero, the radiative transfer equation has an asymptotic behavior which is
governed by a diffusion equation and the corresponding boundary condition.
Generally, a numerical scheme for solving this problem has the truncation error
containing an contribution, that leads to a nonuniform
convergence for small . Such phenomenons require high resolutions
of discretizations, which degrades the performance of the numerical scheme in
the diffusion limit. In this paper, we first provide a--priori estimates for
the scaled spherical harmonic () radiative transfer equation. Then we
present an error analysis for the spherical harmonic discontinuous Galerkin
(DG) method of the scaled radiative transfer equation showing that, under some
mild assumptions, its solutions converge uniformly in to the
solution of the scaled radiative transfer equation. We further present an
optimal convergence result for the DG method with the upwind flux on Cartesian
grids. Error estimates of
(where is the maximum element length) are obtained when tensor product
polynomials of degree at most are used
Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise
One- and multi-dimensional stochastic Maxwell equations with additive noise
are considered in this paper. It is known that such system can be written in
the multi-symplectic structure, and the stochastic energy increases linearly in
time. High order discontinuous Galerkin methods are designed for the stochastic
Maxwell equations with additive noise, and we show that the proposed methods
satisfy the discrete form of the stochastic energy linear growth property and
preserve the multi-symplectic structure on the discrete level. Optimal error
estimate of the semi-discrete DG method is also analyzed. The fully discrete
methods are obtained by coupling with symplectic temporal discretizations. One-
and two-dimensional numerical results are provided to demonstrate the
performance of the proposed methods, and optimal error estimates and linear
growth of the discrete energy can be observed for all cases
Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise
In this paper we propose and analyze finite element discontinuous Galerkin
methods for the one- and two-dimensional stochastic Maxwell equations with
multiplicative noise. The discrete energy law of the semi-discrete DG methods
were studied. Optimal error estimate of the semi-discrete method is obtained
for the one-dimensional case, and the two-dimensional case on both rectangular
meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0
scheme is used as the temporal discretization. Both one- and two-dimensional
numerical results are presented to validate the theoretical analysis results
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