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1. A local discontinuous Galerkin method for nonlinear parabolic SPDEs

   https://doi.org/10.1051/m2an/2020026  

   Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method. 

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2. Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

   https://doi.org/10.1093/imanum/drae094  

   Imbert-Gérard, Lise-Marie; Moiola, Andrea; Perinati, Chiara; Stocker, Paul (January 2025, IMA Journal of Numerical Analysis)

   Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems. 

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3. Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

   https://doi.org/10.1002/num.23126  

   Lu, Jianfang; Jiang, Yan; Shu, Chi‐Wang; Zhang, Mengping (November 2024, Numerical Methods for Partial Differential Equations)

   Abstract In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results. 

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4. Asymptotic Errors in the Superconvergence of Discontinuous Galerkin Methods Based on Upwind-Biased Fluxes for 1D Linear Hyperbolic Equations

   https://doi.org/10.1007/s44007-024-00093-2  

   Lu, Tianshi; Rahmati, Sirvan (March 2024, La Matematica)

   Abstract In this paper, we study the superconvergence of the semi-discrete discontinuous Galerkin (DG) method for linear hyperbolic equations in one spatial dimension. The asymptotic errors in cell averages, downwind point values, and the postprocessed solution are derived for the initial discretization by Gaussian projection (for periodic boundary condition) or Cao projection Cao et al. (SIAM J. Numer. Anal.5, 2555–2573 (2014)) (for Dirichlet boundary condition). We proved that the error constant in the superconvergence of order$$2k+1$$ $2k+1$for DG methods based on upwind-biased fluxes depends on the parity of the orderk. The asymptotic errors are demonstrated by various numerical experiments for scalar and vector hyperbolic equations. 

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5. Central discontinuous Galerkin methods on overlapping meshes for wave equations

   https://doi.org/10.1051/m2an/2020069  

   Liu, Yong; Lu, Jianfang; Shu, Chi-Wang; Zhang, Mengping (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise P k elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k  ≥ 0. In particular, we adopt the techniques in Liu et al . ( SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P 1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results. 

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