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NASA NTRS1989

Calculation of nonequilibrium hydrogen-air reactions with implicit flux vector splitting method

Two methods, fully- and loosely-coupled, are developed to incorporate nonequilibrium hydrogen-air chemistry into the fluid dynamic implicit flux vector splitting code (F3D). The new code (F3D/Chem) is validated against other existing codes for two cases: nozzle expansion, and shock-induced combustion around a blunt body. The shock-induced combustion case is compared also with an experimental data. The reaction rate constants are varied in an effort to reproduce the experimental data. The fully- and loosely-coupled methods are found to yield comparable results, but the computation time is shorter using the loosely-coupled method. The present method is found to reproduce results obtained using different existing codes. The experimental data was not reproduced with any selected rate coefficients set.

Lee, Seung-Ho↗

NASA NTRS1995

The Method of Space-time Conservation Element and Solution Element: Development of a New Implicit Solver

The method of space-time conservation element and solution element is a nontraditional numerical method designed from a physicist's perspective, i.e., its development is based more on physics than numerics. It uses only the simplest approximation techniques and yet is capable of generating nearly perfect solutions for a 2-D shock reflection problem used by Helen Yee and others. In addition to providing an overall view of the new method, we introduce a new concept in the design of implicit schemes, and use it to construct a highly accurate solver for a convection-diffusion equation. It is shown that, in the inviscid case, this new scheme becomes explicit and its amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, its principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme.

Chang, S. C.↗

DOE OSTI2020

Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods

In this study, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1981

Multigrid simulation of asymptotic curved-duct flows using a semi-implicit numerical technique

Asymptotic flows inside curved ducts of rectangular as well as polar cross section are analyzed using the Navier-Stokes equations in terms of the axial velocity and vorticity and the cross-flow stream function. Numerical solutions of the three second-order coupled elliptic partial differential equations governing this flow are obtained efficiently using the coupled alternating-direction implicit (ADI) method as well as the multigrid strongly-implicit (SI) scheme. For the flow configuration studied, the ADI method is found to be more sensitive to the time steps used than is the SI scheme. Use of the multigrid-coupled-strongly-implicit (MG-SI) scheme makes it possible to efficiently obtain fine-grid solutions for configurations having strong secondary flow. It is shown that, for this asymptotic curved-duct flow, the similarity parameter of significance is the Dean's number K rather than the Reynolds number Re. Results are obtained for curved ducts with square cross sections for K up to 900, which here corresponds to Re = 9,000 for this internal flow configuration.

Ghia, K. N.↗

NASA NTRS2024

Intrastep, Stage-Value Predictors for Diagonally-Implicit Runge-Kutta Methods

To better identify the necessary attributes of good stage-value predictors (SVPs), numerous SVPs are designed for an existing: ESDIRK4(3)7L[2]SA [26] and a new: ESDIRK4(3)8L[2]SA scheme.1 Both are stifflyaccurate, stage-order two, explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) schemes. Tradeoffs are studied in the parameter spaces enforcing the constraints on accuracy, linear stability, nonlinear stability and coefficient size to determine which objectives correlate with effective predictors. The SVPs are tested on three challenging external aerodynamics problems [107 − 108 degrees of freedom (DoFs)], each with a different level of stiffness. The problems include two 3D airfoils simulations and one canonical turbulence simulation. All simulations use the compressible Navier-Stokes equations (CNSE). An entropy stable spectral collocation formulation is used for discretizing the spatial terms in the equations. Simulations are performed at a wide variety of temporal error tolerances. Problems that are sufficiently stiff (e.g., lax temporal error tolerances) benefit from SVPs designed with second-order accuracy and stability properties: A-stability, and L-stability, rather than high accuracy constraints. Simulations with modest stiffness (e.g., strict error tolerances) are better suited for SVPs designed using high accuracy constraints. Designing SVPs with enhanced stability properties is tedious but worthwhile. Simulation times are reduced with optimal SVPs by as much as 100% on some stages, with combined stepwise improvements of between 50 − 100% for both methods. A comparative study is performed with the two aforementioned methods as well as four other ESDIRKs. The newly designed ESDIRK4(3)8L[2]SA with γ ≈ 1/10, proves to be the most efficient of the six tested ESDIRK schemes simulating the CNSE.

Diagonally-Implicit Runge-Kutta↗

NASA NTRS2013

High-Order Space-Time Methods for Conservation Laws

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

Huynh, H. T.↗

NASA NTRS1985

Two dimensional and axisymmetric heat transfer results with the CSCM upwind implicit algorithm

The single level conservative supra characteristic method (CSCM-S) is an upwind implicit method that is very robust and in a single level relaxation format makes efficient use of computer resources, and is very rapidly convergent. These factors render the unconditionally stable algorithm a useful compromise between Parabolized Navier-Stokes methods and two level linearized implicit time dependent methods and, thus, the method is very attractive for multidimensional Navier-Stokes problems. In this paper the heat transfer prediction capabilities of the CSCM schemes for flat plate, nozzle, blunt cone and bluff body flows are reported for the first time. Results are compared with available experimental heat transfer measurements.

Bardina, J.↗

NASA NTRS1981

Numerical method for gas dynamics combining characteristic and conservation concepts

An efficient implicit numerical method that solves the compressible Navier-Stokes equations in arbitrary curvilinear coordinates by the finite-volume technique is presented. An intrinsically dissipative difference scheme and a fully implicit treatment of boundary conditions, based on characteristic and conservation concepts, are used to improve stability and accuracy. Efficiency is achieved by using a diagonal form of the implicit algorithm and spatially varying time-steps. Comparisons of various schemes and methods are presented for one- and two-dimensional flows, including transonic separated flow past a thick circular-arc airfoil in a channel. The new method is equal to or better than a version of MacCormack's hybrid method in accuracy and it converges to a steady state up to an order of magnitude faster.

Coakley, T. J.↗

DOE OSTI2021

Verification of a fully implicit particle-in-cell method for the  v ∥ -formalism of electromagnetic gyrokinetics in the XGC code

A fully implicit particle-in-cell method for handling the v ∥ -formalism of electromagnetic gyrokinetics has been implemented in XGC. By choosing the v ∥ -formalism, here we avoid introducing the nonphysical skin terms in Ampère's law, which are responsible for the well-known “cancellation problem” in the p ∥ -formalism. The v ∥ -formalism, however, is known to suffer from a numerical instability when explicit time integration schemes are used due to the appearance of a time derivative in the particle equations of motion from the inductive component of the electric field. Here, using the conventional δf scheme, we demonstrate that our implicitly discretized algorithm can provide numerically stable simulation results with accurate dispersive properties. We verify the algorithm using a test case for shear Alfvén wave propagation in addition to a case demonstrating the ion temperature gradient-kinetic ballooning mode (ITG-KBM) transition. The ITG-KBM transition case is compared to results obtained from other δf gyrokinetic codes/schemes, whose verification has already been archived in the literature.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1986

Report on the feasibility of hypercube concurrent processing systems in computational fluid dynamics

The feasibility of using hypercube-connected concurrent processor systems for problems in computational fluid dynamics is studied. Both explicit and implicit numerical methods are considered and several alternative implementations of these methods are evaluated on concurrent processor systems. A Lax-Wendroff explicit method was designed and implemented for the Navier-Stokes equations. The code runs on the Intel iPSC concurrent processor system. Tests of this code show that it is reasonably efficient. The Beam and Warming implicit factored method was designed and implemented for Berger's equation. Preliminary tests show that the efficiency of code is poor.

Bruno, J.↗

NASA NTRS1985

Implicit finite-difference methods for the Euler equations

The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

Pulliam, T. H.↗

NASA NTRS1982

ADI on staggered mesh - A method for the calculation of compressible convection

An alternating direction implicit (ADI) method has been applied to a staggered grid for the computation of convection in a highly stratified fluid. Since artificial viscosity is not needed, subtle effects like the onset of convection can be studied. These computations compare well with the 2-D results by Graham (1975) and also agree with standard Boussinesq results when taken to that limit. Good efficiency has been achieved with a time step hundreds of times larger than the stability limit imposed by the explicit treatment of diffusion and the Courant number is not restricted to be below 1. The Navier-Stokes equation contains cross spatial derivatives which are treated explicitly in most ADI schemes. The destabilizing effect of such a practice on a 2-D model system with second-order spatial derivative terms only was analyzed and found to be not excessive.

Chan, K. L.↗

NASA NTRS1989

A time accurate finite volume method for propulsion chamber flows

An implicit three-dimensional time-accurate method for propulsion chamber flows is proposed which uses line Gauss-Seidel relaxation and multiple axial sweeps for the convergence of each time step. The general time-integration algorithm employed includes such schemes as the Euler implicit method. The results of spatial and temporal accuracy tests reveal that Roe's (1981) flux difference splitting provides excellent tracking of acoustic wave speeds. In comparison with other methods, no low mean flow Mach number convergence limitation or Courant number stabilization restriction is observed.

Beddini, R. A.↗

DOE OSTI2022

Fast nonlinear iterative solver for an implicit, energy-conserving, asymptotic-preserving charged-particle orbit integrator

Here recently, an asymptotic-preserving (AP) particle orbit integrator has been proposed with remarkable properties including exact energy conservation, the ability to capture of all first-order drifts (including the ∇B-drift), the ability to capture trapped-passing boundaries with parallel velocity extremely close to the critical velocity, and the ability to transition from strongly to weakly magnetized spatial regions. The new AP orbit integrator is implicit, employing a Crank-Nicolson (CN) temporal discretization to ensure exact energy conservation. This, in turn, requires a local nonlinear iteration involving particle velocities and positions, and the local electromagnetic fields, to obtain the new-time solution. Ref. [1] did not attempt to provide an efficient solver for this system, and employed a brute-force GMRES-driven Jacobian-free Newton-Krylov (JFNK) solver to invert the particle orbit equations at every timestep for expediency. While JFNK is robust and reliable, it is also expensive and very intrusive for practical implementations of the method (it requires having the JFNK machinery available and solving a 6 x 6 Jacobian system iteratively once per iteration per particle).

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022