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1. Simple and efficient continuous data assimilation of evolution equations via algebraic nudging

   https://doi.org/10.1002/num.22751  

   Rebholz, Leo G. ; Zerfas, Camille ( January 2021 , Numerical Methods for Partial Differential Equations)

   We introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of theL²projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier–Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long‐time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems.

    

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2. Implementation of the Accurate Conservative Phase Field Method for two-phase incompressible flows in a finite volume framework, Proc. of the 9th Int. and 49th National Conf. on Fluid Mechanics and Fluid Power (FMFP), December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India. FMFP2022–1300.

   Jain, S. ; Tafti, D. K. ( December 2022 , Proc. of the 9th Int. and 49th National Conf. on Fluid Mechanics and Fluid Power (FMFP))

   The phase field method provides a simple mass conserving method for solving two-phase immiscible - incompressible Navier-Stokes Equations. The relative ease in implementing this method compared to other interface reconstruction methods, coupled with its conservativeness and boundedness makes it an attractive alternative. We implement the method in a parallel structured multi-block generalized coordinate finite volume solver using a collocated grid arrangement within the framework of the fractional-step method. The discretization uses a second-order central difference method for both the Navier-Stokes and the phase field equations. A TVD-based averaging technique is used for calculating density at cell faces in the pressure correction step to handle high-density ratios. The simulation framework is verified in standard test cases: Zalesak Disk, a droplet in shear flow, Solitary Wave Runup, Rayleigh Taylor Instability, and the Dam Break Problem. A second-order rate of convergence and excellent phase volume conservation is observed. 

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3. Robust low‐dissipative scheme for curvilinear grids

   https://doi.org/10.1002/fld.4941  

   Arshed, Ghulam M. ; Khan, Ovais U. ( December 2020 , International Journal for Numerical Methods in Fluids)

   This work describes a detailed mathematical procedure in relation to a novel third‐order WENO scheme for the inviscid term of a system of nonlinear equations in the generalized grid system. The scheme developed minimizes the linear and nonlinear sources of dissipation error associated with the classical fifth‐order WENO scheme. The former is minimized by optimizing the resolving efficiency of the scheme whereas the latter is minimized by fixing the accuracy at the second‐order critical point via redefining the nonlinear weights. Moreover, the spectral property of second‐order viscous derivative, approximated by the single and double applications of the standard fourth‐order central finite difference scheme, is presented. The two‐dimensional Euler and Navier–Stokes equations in the generalized grids are mainly pursued. For the robustness in terms of capturing discontinuous and smooth structures, particularly two problems, which are difficult to handle in Cartesian grids, are chosen for discussion. The first one deals with a supersonic shock hitting the circular cylinder and generating all the possible flow inconsistencies. The other one deals with a subsonic flow over a circular cylinder at the incompressible limit. The numerical results are found to be in good agreement with the experimental data.

    

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4. On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case

   https://doi.org/10.1142/S0218202520500438  

   Li, Xiaoli ; Shen, Jie ( November 2020 , Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme. 

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5. Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

   https://doi.org/10.1002/num.22831  

   Layton, William ; Pei, Wenlong ; Qin, Yi ; Trenchea, Catalin ( September 2021 , Numerical Methods for Partial Differential Equations)

   The one‐leg, two‐step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G‐stable. G‐stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier–Stokes equations (NSEs). In this report, we analyze the method of Dahlquist, Liniger and Nevanlinna as a variable step, time discretization of the Navier–Stokes equations. We prove that the kinetic energy is bounded for variable time‐steps, show that the method is second‐order accurate, characterize its numerical dissipation and prove error estimates. The theoretical results are illustrated by several numerical tests.

    

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