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1. Artificial Compressibility Method for the Incompressible Navier–Stokes Equations With Variable Density

   https://doi.org/10.1002/num.70033  

   Cappanera, Loic; Giordano, Salvatore (September 2025, Numerical Methods for Partial Differential Equations)

   ABSTRACT We introduce a novel artificial compressibility technique to approximate the incompressible Navier–Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed method uses the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a method with time‐independent stiffness matrices that is suitable for pseudo‐spectral methods. The stability and temporal convergence of a semi‐implicit version of the method are established under the hypothesis that the density is approximated with a method that conserves the minimum‐maximum principle. Numerical illustrations confirm that both the semi‐implicit and explicit methods are stable and converge with order one under the classic CFL condition. Moreover, the proposed method is shown to perform better than a momentum‐based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi‐fluids in a cylinder. 

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2. Efficient Variable Time-stepping Adaptive DLN Algorithms for the Allen-Cahn Equation

   https://doi.org/10.1007/s10915-025-02980-4  

   Chen, Yiming; Luo, Dianlun; Pei, Wenlong; Xing, Yulong (August 2025, Journal of Scientific Computing)

   Abstract We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditionally non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time-adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time-adaptive DLN methods. 

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3. Advancing parabolic operators in thermodynamic MHD models II: Evaluating a Practical Time Step Limit for Unconditionally Stable Methods

   https://doi.org/10.1088/1742-6596/2742/1/012020  

   Caplan, Ronald M; Johnston, Craig D; Daldoff, Lars_K S; Linker, Jon A (April 2024, Journal of Physics: Conference Series)

   Abstract Unconditionally stable time stepping schemes are useful and often practically necessary for advancing parabolic operators in multi-scale systems. However, serious accuracy problems may emerge when taking time steps that far exceed the explicit stability limits. In our previous work, we compared the accuracy and performance of advancing parabolic operators in a thermodynamic MHD model using an implicit method and an explicit super time-stepping (STS) method. We found that while the STS method outperformed the implicit one with overall good results, it was not able to damp oscillatory behavior in the solution efficiently, hindering its practical use. In this follow-up work, we evaluate an easy-to-implement method for selecting a practical time step limit (PTL) for unconditionally stable schemes. This time step is used to ‘cycle’ the operator-split thermal conduction and viscosity parabolic operators. We test the new time step with both an implicit and STS scheme for accuracy, performance, and scaling. We find that, for our test cases here, the PTL dramatically improves the STS solution, matching or improving the solution of the original implicit scheme, while retaining most of its performance and scaling advantages. The PTL shows promise to allow more accurate use of unconditionally stable schemes for parabolic operators and reliable use of STS methods. 

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4. A Novel Splitting Method for Vlasov-Ampère

   https://doi.org/10.1007/978-3-032-00094-1_43  

   Rossmanith, James A; Vaughan, Christine (February 2026, Springer Nature Switzerland)

   Grabe, M; Oblapenko, G; Torrilhon, M (Ed.)

   Abstract Vlasov equations model the dynamics of plasma in the collisionless regime. A standard approach for numerically solving the Vlasov equation is to operator split the spatial and velocity derivative terms, allowing simpler time-stepping schemes to be applied to each piece separately (known as the Cheng-Knorr method). One disadvantage of such an operator split method is that the order of accuracy of fluid moments (e.g., mass, momentum, and energy) is restricted by the order of the operator splitting (second-order accuracy in the Cheng-Knorr case). In this work, we develop a novel approach that first represents the particle density function on a velocity mesh with a local fluid approximation in each discrete velocity band and then introduces an operator splitting that splits the inter-velocity band coupling terms from the dynamics within the discrete velocity band. The advantage is that the inter-velocity band coupling terms are only needed to achieve consistency of the full distribution functions, but the local fluid models within each band are sufficient to achieve high-order accuracy on global moments such as mass, momentum, and energy. The resulting scheme is verified on several standard Vlasov-Poisson test cases. 

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5. High-order accurate implicit-explicit time-stepping schemes for wave equations on overset grids

   https://doi.org/10.1016/j.jcp.2024.113513  

   Carson, Allison M; Banks, Jeffrey W; Henshaw, William D; Schwendeman, Donald W (January 2025, Journal of Computational Physics)

   New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids. The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations. 

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