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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. Primal Dual Methods for Wasserstein Gradient Flows

   https://doi.org/10.1007/s10208-021-09503-1  

   Carrillo, José A.; Craig, Katy; Wang, Li; Wei, Chaozhen (March 2021, Foundations of Computational Mathematics)

   null (Ed.)

   Abstract Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method. 

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3. Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation

   https://doi.org/10.1051/m2an/2021020  

   Guo, Hui; Jia, Rui; Tian, Lulu; Yang, Yang (May 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we apply two fully-discrete local discontinuous Galerkin (LDG) methods to the compressible wormhole propagation. We will prove the stability and error estimates of the schemes. Traditional LDG methods use the diffusion term to control of convection term to obtain the stability for some linear equations. However, the variables in wormhole propagation are coupled together and the whole system is highly nonlinear. Therefore, it is extremely difficult to obtain the stability for fully-discrete LDG methods. To fix this gap, we introduce a new auxiliary variable including both the convection and diffusion terms. Moreover, we also construct a special time integration for the porosity, leading to physically relevant numerical approximations and controllable growth rate of the porosity. With a reasonable growth rate, it is possible to handle the time level mismatch in the first-order fully discrete scheme and obtain the stability of the scheme. For the whole system, we will prove that under weak temporal-spatial conditions, the optimal error estimates for the pressure, velocity, porosity and concentration under different norms can be obtained. Numerical experiments are also given to verify the theoretical results. 

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4. Optimal Order Space‐Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations With Multiplicative Noise

   https://doi.org/10.1002/num.70067  

   Feng, Xiaobing; Li, Yukun; Vo, Liet (January 2026, Numerical Methods for Partial Differential Equations)

   This paper develops and analyzes an optimal‐order semi‐discrete scheme and its fully discrete finite element approximation for a second‐order system of nonlinear stochastic equations with multiplicative noise, which governs wave propagation in elastic media under the influence of random damping and force. A nonstandard time‐stepping scheme is introduced for time discretization. It is shown that the scheme converges with rates and respectively in the energy‐ and norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the finite element method is constructed. It is proven that the fully discrete method converges with optimal rates and in the energy‐ and ‐norm. The crux of the analysis is to establish some high‐moment stability results and utilize a refined error estimate for the trapezoidal quadrature rule to control the nonlinearities from the drift term and the multiplicative noise. Numerical experiments are also provided to validate the theoretical results. 

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5. A space–time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients

   https://doi.org/10.1090/mcom/3786  

   Imbert-Gérard, Lise-Marie; Moiola, Andrea; Stocker, Paul (May 2023, Mathematics of Computation)

   Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise-constant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions. 

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