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1. Christov expansion method for nonlocal nonlinear evolution equations

   https://doi.org/10.1088/1742-6596/2675/1/012022  

   Christou, M A ; Christov, I C ( December 2023 , Journal of Physics: Conference Series)

   Todorov, M D (Ed.)

   Christov functions are a complete orthonormal set of functions on L^2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves. 

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2. Quantum hydrodynamic approximations to the finite temperature trapped Bose gases

   Shi Jin, Minh-Binh Tran ( October 2019 , Physica. D, Nonlinear phenomena)

   For the quantum kinetic system modelling the Bose-Einstein Condensate that accounts for interactions between condensate and excited atoms, we use the Chapman-Enskog expansion to derive its hydrodynamic approximations, include both Euler and Navier-Stokes approximations. The hydrodynamic approximations describe not only the macroscopic behavior of the BEC but also its coupling with the non-condensates, which agrees with the Landau two-fluid theory. 

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3. Multi-discretization domain specific language and code generation for differential equations

   https://doi.org/10.1016/j.jocs.2023.101981  

   Heisler, Eric ; Deshmukh, Aadesh ; Mazumder, Sandip ; Sadayappan, Ponnuswamy ; Sundar, Hari ( April 2023 , Journal of computational science)

   Finch, a domain specific language and code generation framework for partial differential equations (PDEs), is demonstrated here to solve two classical problems: steady-state advection diffusion equation (single PDE) and the phonon Boltzmann transport equation (coupled PDEs). Both finite volume and finite element methods are explored. In addition to work presented at the 2022 International Conference on Computational Science (Heisler et al., 2022), we include recent developments for solving nonlinear equations using both automatic and symbolic differentiation, and demonstrate the capability for the Bratu (nonlinear Poisson) equation. 

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4. Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space

   https://doi.org/10.1007/s00205-021-01651-6  

   Guo, Yan ; Huang, Feimin ; Wang, Yong ( July 2021 , Archive for Rational Mechanics and Analysis)

   Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic study of the viscous layer equations and the L2 to L∞ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations in half-space. 

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5. Optimal error estimates of ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional convection-diffusion and biharmonic equations

   https://doi.org/10.1090/mcom/3927  

   Chen, Yuan ; Xing, Yulong ( September 2024 , Mathematics of Computation)

   In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results.

    

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