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In this paper we propose a simple and unified framework to investigate the L 2-norm stability of the explicit Runge-Kutta discontinuous Galerkin (RKDG) methods, when solving the linear constant-coefficient hyperbolic equations. Two key ingredients in the energy analysis are the temporal differences of numerical solutions in different Runge-Kutta stages, and a matrix transferring process. Many popular schemes, including the fourth order RKDG schemes, are discussed in this paper to show that the presented technique is flexible and useful. Different performances in the L 2-norm stability of different RKDG schemes are carefully investigated. For some lower-degree piecewise polynomials, the monotonicity stability is proved if the stability mechanism can be provided by the upwind-biased numerical fluxes. Some numerical examples are also given.

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Seepage force is simplified as seepage volumetric force in the stress field along the radial direction. Out-of-plane stress and seepage force are incorporated, and the theoretical solutions for stress, displacement, and plastic radius of a circular opening for the elastic-brittle-plastic and elastic-plastic rock mass are proposed based on the Mohr-Coulomb (MC) and generalized Hoek-Brown (HB) failure criteria. The presented solution and Wang's solution (2012) are compared, and the corrected version of the proposed method is validated. Numerical examples of the proposed method based on the MC and generalized HB failure criteria reveal that the distributions of stress and displacement in the surrounding rock of the tunnel are significantly influenced by seepage force and out-ofplane stress. Displacement and plastic radius when seepage force and out-of-plane stress are considered are larger than those when the seepage force is not considered; the regulations of stress, however, run opposite. The results of displacement and plastic radius based on the generalized HB failure criterion are larger than those based on the MC failure criterion.

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We construct K-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions u of the equation such thatwhere K ≥ 2 and for any k ∈ {1, . . . , K}, W k is the Lorentz transform of the explicit standing soliton W (x) = (1 + |x| 2 /15) −3/2 , with any speed ℓ k ∈ R 5 , |ℓ k | < 1 satisfying ℓ k ′ = ℓ k for k ′ = k, and an explicit smallness condition. The proof extends the refined method of construction of asymptotic multi-solitons from [11,12].

In order to investigate the influence of the intermediate principal stress on the stress and displacement of surrounding rock, a novel approach based on 3D Hoek-Brown (H-B) failure criterion was proposed. Taking the strain-softening characteristic of rock mass into account, the potential plastic zone is subdivided into a finite number of concentric annulus and a numerical procedure for calculating the stress and displacement of each annulus was presented. Strains were obtained based on the nonassociated and associated flow rule and 3D plastic potential function. Stresses were achieved by the stress equilibrium equation and generalized Hoek-Brown failure criterion. Using the proposed approach, we can get the solutions of the stress and displacement of the surrounding rock considering the intermediate principal stress. Moreover, the proposed approach was validated with the published results. Compared with the results based on generalized Hoek-Brown failure criterion, it is shown that the plastic radius calculated by 3D Hoek-Brown failure criterion is smaller than those solved by generalized H-B failure criterion, and the influences of dilatancy effect on the results based on the generalized H-B failure criterion are greater than those based on 3D H-B failure criterion. The displacements considering the nonassociated flow rule are smaller than those considering associated flow rules.