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Qiang Zhang - One of the best experts on this subject based on the ideXlab platform.
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Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for linear hyperbolic equation in one-dimension with discontinuous Initial data
2020Co-Authors: Qiang Zhang, Chi-wang ShuAbstract:Abstract. In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve linear hyperbolic equation in one dimension with discontinuous but piecewise smooth Initial data. The discontinuous finite element space is made up of piecewise polynomials of arbitrary degree, and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The error at the final time T in the L 2 (R\R T )-norm is the optimal order both in space and in time, where R T is the pollution region due to the Initial Discontinuity with the width of order O(h 1/2 log(1/h)), where h is the maximum cell length
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error estimates for the third order explicit runge kutta discontinuous galerkin method for a linear hyperbolic equation in one dimension with discontinuous Initial data
Numerische Mathematik, 2014Co-Authors: Qiang ZhangAbstract:In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve a linear hyperbolic equation in one dimension with discontinuous but piecewise smooth Initial data. The discontinuous finite element space is made up of piecewise polynomials of arbitrary degree $$k\ge 1$$ k ? 1 , and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The $$L^2(\mathbb R \backslash \mathcal R _T)$$ L 2 ( R ? R T ) -norm error at the final time $$T$$ T is optimal in both space and time, where $$\mathcal R _T$$ R T is the pollution region due to the Initial Discontinuity with the width $$\mathcal O (\sqrt{T\beta }h^{1/2}\log (1/h))$$ O ( T β h 1 / 2 log ( 1 / h ) ) . Here $$h$$ h is the maximum cell length and $$\beta $$ β is the flowing speed. These results are independent of the time step and hold also for the semi-discrete discontinuous Galerkin method.
Jean-françois Coulombel - One of the best experts on this subject based on the ideXlab platform.
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Weakly nonlinear surface waves in magnetohydrodynamics
Asymptotic Analysis, 2020Co-Authors: Olivier Pierre, Jean-françois CoulombelAbstract:This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the incompressible magnetohydrodynamics equations. Current vortex sheets are piecewise smooth solutions to the incompressible magnetohydrodynamics equations that satisfy suitable jump conditions for the velocity and magnetic field on the (free) Discontinuity surface. In this work, we complete an earlier work by Ali and Hunter and construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the Initial Discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to `surface waves' on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobi type equation known as the `HIZ equation' (standing for Hamilton-Il'insky-Zabolotskaya) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. Based on a suitable duality formula, we exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. The verification of these solvability conditions is based on a combination of mere algebra and arguments of combinatorial analysis. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the free boundary problem. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves does not arise in the same way for the current vortex sheet problem.
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A priori Estimates for 3D Incompressible Current-Vortex Sheets
Communications in Mathematical Physics, 2011Co-Authors: Jean-françois Coulombel, Alessandro Morando, Paolo Secchi, Paola TrebeschiAbstract:We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the Initial Discontinuity and a flatness condition on the Initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.
Paola Trebeschi - One of the best experts on this subject based on the ideXlab platform.
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well posedness of the linearized plasma vacuum interface problem in ideal incompressible mhd
Quarterly of Applied Mathematics, 2014Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola TrebeschiAbstract:We consider the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the Initial Discontinuity, the well-posedness of the linearized problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard Initial-boundary value problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized problem. For the regularized problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .
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A priori Estimates for 3D Incompressible Current-Vortex Sheets
Communications in Mathematical Physics, 2011Co-Authors: Jean-françois Coulombel, Alessandro Morando, Paolo Secchi, Paola TrebeschiAbstract:We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the Initial Discontinuity and a flatness condition on the Initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.
Chi-wang Shu - One of the best experts on this subject based on the ideXlab platform.
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Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for linear hyperbolic equation in one-dimension with discontinuous Initial data
2020Co-Authors: Qiang Zhang, Chi-wang ShuAbstract:Abstract. In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve linear hyperbolic equation in one dimension with discontinuous but piecewise smooth Initial data. The discontinuous finite element space is made up of piecewise polynomials of arbitrary degree, and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The error at the final time T in the L 2 (R\R T )-norm is the optimal order both in space and in time, where R T is the pollution region due to the Initial Discontinuity with the width of order O(h 1/2 log(1/h)), where h is the maximum cell length
Yuri Trakhinin - One of the best experts on this subject based on the ideXlab platform.
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well posedness of the linearized plasma vacuum interface problem in ideal incompressible mhd
Quarterly of Applied Mathematics, 2014Co-Authors: Alessandro Morando, Yuri Trakhinin, Paola TrebeschiAbstract:We consider the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the Initial Discontinuity, the well-posedness of the linearized problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard Initial-boundary value problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized problem. For the regularized problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .
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The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics
Archive for Rational Mechanics and Analysis, 2009Co-Authors: Yuri TrakhininAbstract:We prove the local-in-time existence of solutions with a surface of current-vortex sheet (tangential Discontinuity) of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the Initial Discontinuity. This paper is a natural completion of our previous analysis ( Trakhinin in Arch Ration Mech Anal 177:331–366, 2005) where a sufficient condition for the weak stability of planar current-vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities. The original nonlinear problem is a free boundary hyperbolic problem. Since the free boundary is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces $${H^m_*}$$ . The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. Therefore, we prove our existence theorem by a suitable Nash–Moser-type iteration scheme.