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1. Von Neumann Stable, Implicit, High Order, Finite Volume WENO Schemes

   https://doi.org/10.2118/193817-MS  

   Arbogast, Todd ; Huang, Chieh-Sen ; Zhao, Xikai ( April 2019 , SPE Reservoir Simulation Conference 2019)

   Simulation of flow and transport in petroleum reservoirs involves solving coupled systems of advection-diffusion-reaction equations with nonlinear flux functions, diffusion coefficients, and reactions/wells. It is important to develop numerical schemes that can approximate all three processes at once, and to high order, so that the physics can be well resolved. In this paper, we propose an approach based on high order, finite volume, implicit, Weighted Essentially NonOscillatory (iWENO) schemes. The resulting schemes are locally mass conservative and, being implicit, suited to systems of advection-diffusion-reaction equations. Moreover, our approach gives unconditionally L-stable schemes for smooth solutions to the linear advection-diffusion-reaction equation in the sense of a von Neumann stability analysis. To illustrate our approach, we develop a third order iWENO scheme for the saturation equation of two-phase flow in porous media in two space dimensions. The keys to high order accuracy are to use WENO reconstruction in space (which handles shocks and steep fronts) combined with a two-stage Radau-IIA Runge-Kutta time integrator. The saturation is approximated by its averages over the mesh elements at the current time level and at two future time levels; therefore, the scheme uses two unknowns per grid block per variable, independent of the spatial dimension. This makes the scheme fairly computationally efficient, both because reconstructions make use of local information that can fit in cache memory, and because the global system has about as small a number of degrees of freedom as possible. The scheme is relatively simple to implement, high order accurate, maintains local mass conservation, applies to general computational meshes, and appears to be robust. Preliminary computational tests show the potential of the scheme to handle advection-diffusion-reaction processes on meshes of quadrilateral gridblocks, and to do so to high order accuracy using relatively long time steps. The new scheme can be viewed as a generalization of standard cell-centered finite volume (or finite difference) methods. It achieves high order in both space and time, and it incorporates WENO slope limiting. 

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2. Semi-implicit hybrid discrete (H$^T_N$) approximation of thermal radiative transfer

   https://doi.org/https://link.springer.com/article/10.1007/s10915-021-01686-7  

   R.G. McClarren, J.A. Rossmanith ( November 2021 , Journal of scientific computing)

   The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$_N$ (spherical harmonics) or the S$_N$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$^T_N$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$_N$ and S$_N$, and indeed reduces to each of these approximations in various limits: H$^1_N$ $\equiv$ P$_N$ and H$^T_0$ $\equiv$ S$_T$. We prove that H$^T_N$ results in a system of hyperbolic partial differential equations for all $T\ge 1$ and $N\ge 0$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop a numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H$^T_N$ ansatz and the numerical discretizations. 

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3. A positivity‐preserving and energy stable scheme for a quantum diffusion equation

   https://doi.org/10.1002/num.22809  

   Huo, Xiaokai ; Liu, Hailiang ( July 2021 , Numerical Methods for Partial Differential Equations)

   We propose a new fully‐discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully‐discretized scheme with proven positivity‐preserving and energy stable properties using only standard finite difference discretization. The difficulty in proving the positivity‐preserving property lies in the lack of a maximum principle for fourth order partial differential equations. To overcome this difficulty, we reformulate the scheme as an optimization problem based on a variational structure and use the singular nature of the energy functional near the boundary values to exclude the possibility of non‐positive solutions. The scheme is also shown to be mass conservative and consistent.

    

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4. A fast local embedded boundary method suitable for high power electromagnetic sources

   https://doi.org/10.1063/5.0019210  

   Thavappiragasam, Mathialakan ; Christlieb, Andrew ; Luginsland, John ; Guthrey, Pierson ( November 2020 , AIP Advances)

   High power sources of electromagnetic energy often require complicated structures to support electromagnetic modes and shape electromagnetic fields to maximize the coupling of the field energy to intense relativistic electron beams. Geometric fidelity is critical to the accurate simulation of these High Power Electromagnetic (HPEM) sources. Here, we present a fast and geometrically flexible approach to calculate the solution to Maxwell’s equations in vector potential form under the Lorenz gauge. The scheme is an implicit, linear-time, high-order, A-stable method that is based on the method of lines transpose (MOLT). As presented, the method is fourth order in time and second order in space, but the A-stable formulation could be extended to both high order in time and space. An O(n) fast convolution is employed for space-integration. The main focus of this work is to develop an approach to impose perfectly electrically conducting (PEC) boundary conditions in MOLT by extending our past work on embedded boundary methods. As the method is A-stable, it does not suffer from small time step limitations that are found in explicit finite difference time domain methods when using either embedded boundary or cut-cell methods to capture geometry. This is a major advance for the simulation of HPEM devices. While there is no conceptual limitation to develop this in 3D, our initial work has centered on 2D. The extension to 3D requires validation that the proposed fixed point iteration will converge and is the subject of our follow-up work. The eventual goal is to combine this method with particle methods for the simulations of plasma. In the current work, the scheme is evaluated for EM wave propagation within an object that is bounded by PEC. The consistency and performance of the scheme are confirmed using the ping test and frequency mode analysis for rotated square cavities—a standard test in the HPEM community. We then demonstrate the diffraction Q value test and the use of this method for simulating an A6 magnetron. The ability to handle both PEC and open boundaries in a standard device test problem, such as the A6, gives confidence on the robustness of this new method.

    

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5. A Robust Bubble Growth Solution Scheme for Implementation in CFD Analysis of Multiphase Flows

   https://doi.org/10.3390/computation11040072  

   Pang, Hao ; Ngaile, Gracious ( April 2023 , Computation)

   Although the full form of the Rayleigh–Plesset (RP) equation more accurately depicts the bubble behavior in a cavitating flow than its reduced form, it finds much less application than the latter in the computational fluid dynamic (CFD) simulation due to its high stiffness. The traditional variable time-step scheme for the full form RP equation is difficult to be integrated with the CFD program since it requires a tiny time step at the singularity point for convergence and this step size may be incompatible with time marching of conservation equations. This paper presents two stable and efficient numerical solution schemes based on the finite difference method and Euler method so that the full-form RP equation can be better accepted by the CFD program. By employing a truncation bubble radius to approximate the minimum bubble size in the collapse stage, the proposed schemes solve for the bubble radius and wall velocity in an explicit way. The proposed solution schemes are more robust for a wide range of ambient pressure profiles than the traditional schemes and avoid excessive refinement on the time step at the singularity point. Since the proposed solution scheme can calculate the effects of the second-order term, liquid viscosity, and surface tension on the bubble evolution, it provides a more accurate estimation of the wall velocity for the vaporization or condensation rate, which is widely used in the cavitation model in the CFD simulation. The legitimacy of the solution schemes is manifested by the agreement between the results from these schemes and established ones from the literature. The proposed solution schemes are more robust in face of a wide range of ambient pressure profiles. 

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