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1. Development of a Stable High-Order Point-Value Enhanced Finite Volume (PFV) Method Based on Approximate Delta Functions

   https://doi.org/10.2514/6.2018-4168  

   Xuan, Li-Jun; Majdalani, Joseph (June 2018, 48th AIAA Fluid Dynamics Conference)

   null (Ed.)

   In this work, we generalize the expression of an approximate delta function (ADF), which is a finite- order polynomial that holds identical integral properties to the Dirac delta function, particularly, when used in conjunction with a finite-order polynomial integrand over a finite domain. By focusing on one- dimensional configurations, we show that the use of generalized ADF polynomials can be effective at recovering and extending several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements reduces the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Presently, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to two benchmark cases, namely, the linear wave and nonlinear Burgers’ equations in one-dimensional space. 

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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. An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

   https://doi.org/10.1002/num.22274  

   Jiang, Yi; Liu, Hailiang (May 2018, Numerical Methods for Partial Differential Equations)

   In this article, we introduce an invariant‐region‐preserving (IRP) limiter for thep‐system and the corresponding viscousp‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to thep‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscousp‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes. 

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4. Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

   https://doi.org/10.1063/5.0158146  

   Chen, Kuangxu; Liang, Chunlei; Wan, Minping (August 2023, Physics of Fluids)

   We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets. 

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5. Second order accurate particle-in-cell discretization of the Navier-Stokes equations

   https://doi.org/10.1016/j.jcp.2024.113302  

   Jiang, Han; Schroeder, Craig (December 2024, Journal of Computational Physics)

   We propose the use of PolyPIC transfers [10] to construct a second order accurate discretization of the Navier-Stokes equations within a particle-in-cell framework on MAC grids. We investigate the accuracy of both APIC [16], [17], [8] and quadratic PolyPIC [10] transfers and demonstrate that they are suitable for constructing schemes converging with orders of approximately 1.5 and 2.5 respectively. We combine PolyPIC transfers with BDF-2 time integration and a splitting scheme for pressure and viscosity and demonstrate that the resulting scheme is second order accurate. Prior high order particle-in-cell schemes interpolate accelerations (not velocities) from the grid to particles and rely on moving least squares to transfer particle velocities to the computational grid. The proposed method instead transfers velocities to particles, which avoids the accumulation of noise on particle velocities but requires the polynomial reconstruction to be performed using polynomials that are one degree higher. Since this polynomial reconstruction occurs over the regular grid (rather than irregularly distributed particles), the resulting weighted least squares problem has a fixed sparse structure, can be solved efficiently in closed form, and is independent of particle coverage. 

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