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• Derived a new class of nonlinearly stable flux reconstruction schemes in split form. • Nonlinearly stable flux reconstruction correction functions must be applied on the nonlinear volume terms. • Flux reconstruction in split form with the correction functions only reconstructing the surface is divergent. • The split form must be embedded within the discrete stiffness operator for nonlinear stability. The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted for Burgers' equation verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy. [ABSTRACT FROM AUTHOR]

This paper is devoted to an emerging research topic, which is the numerical computation of stationary states of a generic Dirac operator with nonlinear potential. We are more specifically interested in the numerical computation of chemical potentials and eigenenergies. In this goal, several approaches are explored namely Feit-Fleck's, Rayleigh-Ritz, and min-max methods for the computation of chemical potentials, and normalized gradient flow methods for the eigenenergy computation. Balance operators will be introduced to ensure the convergence of some of the proposed methods. Finally, some numerical experiments will be proposed in order to validate the presented methods. • Derivation and analysis of computational methods for nonlinear Dirac operators. • Balanced methods and convergent direct methods for computing chemical potentials and eigenenergies. • Analysis of variational collapse and other issues due to Dirac operator's unbounded spectrum. • Overview of different methods and strategies. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work [1] where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment. • The proposal of a numeric-symbolic hybrid deep network to recover PDEs from observed dynamic data. • The symbolic network is able to recover concise analytic form of the hidden PDE model. • Our approach only requires minor prior knowledge on the mechanism of the observed dynamic data. • The network can perform accurate long-term prediction without re-training for new initial conditions. [ABSTRACT FROM AUTHOR]