3 research outputs found
Enforcing strong stability of explicit Runge--Kutta methods with superviscosity
A time discretization method is called strongly stable, if the norm of its
numerical solution is nonincreasing. It is known that, even for linear
semi-negative problems, many explicit Runge--Kutta (RK) methods fail to
preserve this property. In this paper, we enforce strong stability by modifying
the method with superviscosity, which is a numerical technique commonly used in
spectral methods. We propose two approaches, the modified method and the
filtering method for stabilization. The modified method is achieved by
modifying the semi-negative operator with a high order superviscosity term; the
filtering method is to post-process the solution by solving a diffusive or
dispersive problem with small superviscosity. For linear problems, most
explicit RK methods can be stabilized with either approach without accuracy
degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on
diffusive superviscosity for ensuring strong stability. The bound we derived
for general dispersive-diffusive superviscosity is also verified to be sharp
numerically. For nonlinear problems, a filtering method is investigated for
stabilization. Numerical examples with linear non-normal ordinary differential
equation systems and for discontinuous Galerkin approximation of conservation
laws are performed to validate our analysis and to test the performance.Comment: 40 page
Error analysis of Runge--Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations
In this paper, we present error estimates of fully discrete Runge--Kutta
discontinuous Galerkin (DG) schemes for linear time-dependent partial
differential equations. The analysis applies to explicit Runge--Kutta time
discretizations of any order. For spatial discretization, a general discrete
operator is considered, which covers various DG methods, such as the
upwind-biased DG method, the central DG method, the local DG method and the
ultra-weak DG method. We obtain error estimates for stable and consistent fully
discrete schemes, if the solution is sufficiently smooth and a spatial operator
with certain properties exists. Applications to schemes for hyperbolic
conservation laws, the heat equation, the dispersive equation and the wave
equation are discussed. In particular, we provide an alternative proof of
optimal error estimates of local DG methods for equations with high order
derivatives in one dimension, which does not rely on energy inequalities of
auxiliary unknowns
On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity
In this paper, we present and study discontinuous Galerkin (DG) methods for
one-dimensional multi-symplectic Hamiltonian partial differential equations. We
particularly focus on semi-discrete schemes with spatial discretization only,
and show that the proposed DG methods can simultaneously preserve the
multi-symplectic structure and energy conservation with a general class of
numerical fluxes, which includes the well-known central and alternating fluxes.
Applications to the wave equation, the Benjamin-Bona-Mahony equation, the
Camassa-Holm equation, the Korteweg-de Vries equation and the nonlinear
Schr\"odinger equation are discussed. Some numerical results are provided to
demonstrate the accuracy and long time behavior of the proposed methods.
Numerically, we observe that certain choices of numerical fluxes in the
discussed class may help achieve better accuracy compared with the commonly
used ones including the central fluxes.Comment: 34 page