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1. Second‐order time discretization for a coupled quasi‐Newtonian fluid‐poroelastic system

   https://doi.org/10.1002/fld.4801  

   Kunwar, Hemanta; Lee, Hyesuk; Seelman, Kyle (December 2019, International Journal for Numerical Methods in Fluids)

   Summary Numerical methods are proposed for the nonlinear Stokes‐Biot system modeling interaction of a free fluid with a poroelastic structure. We discuss time discretization and decoupling schemes that allow the fluid and the poroelastic structure computed independently using a common stress force along the interface. The coupled system of nonlinear Stokes and Biot is formulated as a least‐squares problem with constraints, where the objective functional measures violation of some interface conditions. The local constraints, the Stokes and Biot models, are discretized in time using second‐order schemes. Computational algorithms for the least‐squares problems are discussed and numerical results are provided to compare the accuracy and efficiency of the algorithms. 

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2. Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

   https://doi.org/10.1002/nme.6158  

   Al Mahbub, Md. Abdullah; He, Xiaoming; Nasu, Nasrin Jahan; Qiu, Changxin; Zheng, Haibiao (July 2019, International Journal for Numerical Methods in Engineering)

   Summary In this paper, we propose and analyze two stabilized mixed finite element methods for the dual‐porosity‐Stokes model, which couples the free flow region and microfracture‐matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual‐porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. 

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3. Noniterative localized exponential time differencing methods for hyperbolic conservation laws

   https://doi.org/10.1007/s10444-025-10240-0  

   Doan, Cao-Kha; Huynh, Phuoc-Toan; Hoang, Thi-Thao-Phuong (June 2025, Advances in Computational Mathematics)

   The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization at each time step. The scheme is fully explicit and relies on the computation of matrix exponential vector products. To accelerate such computation, we further construct a noniterative, nonoverlapping domain decomposition algorithm, namely localized ETDRK2, which loosely decouples the system at each time step via suitable interface conditions. Temporal error analysis of the proposed global and localized ETDRK2 schemes is rigorously proved; moreover, the schemes are shown to be conservative under periodic boundary conditions. Numerical results for the Burgers' equation in one and two dimensions (with moving shocks) are presented to verify the theoretical results and illustrate the performance of the global and localized ETDRK2 methods where large time step sizes can be used without affecting numerical stability. 

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4. 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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5. Second‐order, loosely coupled methods for fluid‐poroelastic material interaction

   https://doi.org/10.1002/num.22452  

   Oyekole, Oyekola; Bukač, Martina (December 2019, Numerical Methods for Partial Differential Equations)

   Abstract This work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time‐dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin–Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for the coupled problem. The first method uses the second‐order backward differentiation formula (BDF2) for implicit integration, while treating the interface terms explicitly using a second‐order extrapolation formula. The second method is the Crank–Nicolson and Leap‐Frog (CNLF) method, where the Crank–Nicolson method is used to implicitly advance the solution in time, while the coupling terms are explicitly approximated by the Leap‐Frog integration. We show that the BDF2 method is unconditionally stable and uniformly stable in time, while the CNLF method is stable under a CFL condition. Both schemes are validated using numerical simulations. Second‐order convergence in time is observed for both methods. Simulations over a longer period of time show that the errors in the solution remain bounded. Cases when the structure is poroviscoelastic and poroelastic are included in numerical examples. 

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