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Numerical methods for integrating nonlinear differential equations with parasitic eigenvalues
The A-contractivity of Runge-Kutta methods with respect to an inner product norm was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is extended to contractivity in a region bounded by a circle through the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point. It is shown that an explicit Runge-Kutta method cannot be contractive in any circle of this class if it is more than fourth order accurate.
The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.
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This paper presents a general solution algorithm for the set of difference equations that arise when two-point central differences are used to approximate the flux difference terms in systems of hyperbolic differential equations. The general algorithm eliminates the weak points associated with the nonstandard algorithm reported by Wornom and Hafez (1986). The disadvantages of their algorithm relate to its implementation. It consists of separate algorithms for subsonic, supersonic, sonic and shock cells, applied individually, which presents a major bookkeeping problem when multiple sonic and shock cells are present. The general algorithm eliminates this problem and introduces an improved shock treatment which produces shocks with at most one interior shock point.
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The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) methods (e.g. the leap-frog scheme) for both spatial and temporal integrations. Here, we employ a pseudospectral method for solving the Poisson equation and a fully implicit time integration to achieve exact energy conservation. The advantage of a pseudospectral field solver is its spectral accuracy in solving field solutions. Earlier studies of implicit time integration of PIC FD equations can enforce exact energy exchange between field and particles, resulting in exact energy-conserving schemes. Here, we prove that the exact energy conservation property can be carried over to the pseudospectral scheme. Simultaneously, we provide a solution to ensure a pseudospectral charge continuity equation. We demonstrate the new scheme in a 2D electrostatic PIC code. In conclusion, theoretical results are confirmed via numerical examples.
The mean velocity profile across a fully developed turbulent duct flow is obtained from an eddy viscosity relation combined with an empirical outer region wake function. Results are in good agreement with experiments and with direct numerical simulations in the same flow at two Reynolds numbers. In particular, the near-wall trend of the Reynolds shear stress and its variation with Reynolds number are similar to those of the simulations. The eddy viscosity method is more accurate than previous mixing length or implicit function methods.
The alternating direction implicit (ADI) method is adopted, modified, and applied to the Reynolds equation for thin, gas fluid films. An efficient code is developed to predict both the steady-state and dynamic performance of an aerodynamic journal bearing. An alternative approach is shown for hybrid journal gas bearings by using Liebmann's iterative solution (LIS) for elliptic partial differential equations. The results are compared with known design criteria from experimental data. The developed methods show good accuracy and very short computer running time in comparison with methods based on an inverting of a matrix. The computer codes need a small amount of memory and can be run on either personal computers or on mainframe systems.
A domain decomposition method for implicit schemes that require significantly less storage and is several times faster than factorization algorithms is proposed. The transient domain decomposition method is an extension of the finite element tearing and interconnecting (FETI) method for the solution of static problems. Serial and parallel performance results obtained using the CRAY Y-MP/8 and the iPSC-860/128 systems demonstrate that the FETI method is superior to both serial and parallel direct methods.
In a recently published work by Abarbanel and Gottlieb (1980), a new class of explicit time-split algorithms designed for application to the compressible Navier-Stokes equations was developed. These algorithms, which utilize locally-one-dimensional (LOD) spatial steps, were shown to possess stability characteristics superior to those of other time-split schemes. In the present work, the properties of an implicit LOD method, analogous to the Abarbanel-Gottlieb algorithm, are examined using the two-dimensional heat conduction equation as the test problem. Both temporal and spatial inconsistencies inherent in the scheme are identified, and a new consistent, implicit splitting approach is developed and applied to the linear Burgers' equation. The relationship between this new method and other time-split implicit schemes is explained and stability problems encountered with the method in three dimensions are discussed.
Joule heating has been regarded as an energy-efficient and sustainable method for heating materials and gases at large scales. The modeling of local temperature effects at pore-resolved scales for such systems, however, has been difficult to achieve due to challenges in coupling thermo-chemical processes in complex porous media and in large representative volume elements (RVEs). To this end, we developed an electro-thermal model at the pore scale to study Joule heating effects in large heterogeneous systems with different microstructures. This was achieved using the level set method to implicitly delineate distinct regions within the domain, and an embedded boundary method to facilitate heat exchange across the fluid-solid interface. Moreover, we applied this method to investigate unsteady non-linear electro-thermal effects in non-woven fibrous graphite conductors for RVEs with characteristic lengths of 2 mm, with different fiber orientations, porosity (80% – 90%) and fiber diameters (10 – 20µm). The coupled equations were solved numerically and they produced peak temperatures greater than 2000 K resulting in heating rates as high as 80,000 K/s. Moreover, the results depended strongly on the microstructure of the fiber skeleton and current density. Geometries with large fibers (∼ 20µm) had the highest average and peak temperatures with the mean temperature increasing by 3.9 % while the peak temperature increased by 9.9 %. Anisotropic domains on the other hand had the lowest mean and peak temperatures with peak and mean temperatures of 2293 K and 1437.7K respectively representing a corresponding 12.1% and 5.1% drop in the temperatures. An increase in porosity from 80% to 90%, however, led to an increase in the peak temperature by 5.1%.
In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.
The behavior of fluids at supercritical thermodynamic conditions is inherently complex due to large variations in thermodynamic and transport properties. Recent numerical and experimental investigations illustrate ongoing interest for these fluids, especially supercritical CO 2 and supercritical water, for a variety of applications. For example, supercritical water reactors (SCWR) operate in this extreme condition of high-pressure and temperature, resulting in highly dynamic flow fields and unexpected heat transfer regimes. The potential heat transfer benefits in this regime are directly associated with the extreme variations in thermodynamic and transport properties, which occur at, and above, the critical point. This work characterizes the hydrodynamic instabilities that arise for fluids at supercritical thermodynamic conditions when buoyancy forces are significant. Two specific configurations are considered, a natural convection cavity flow, and a mixed convection, heated, horizontal channel flow. Natural convection flow in a cavity is a classical configuration with expected behavior below the critical point. This configuration aids in characterizing the effect of the variable properties in the supercritical thermodynamic regime. Further, limited studies in the existing literature have been conducted for low-Reynolds and intermediate-Rayleigh numbers, mixed-convection channel flows for supercritical water, which is the focus of the channel flow configuration. To investigate the thermally driven hydrodynamic instabilities in this regime, a high-order fully-implicit numerical method is used. Such strong variations in thermophysical properties (in particular, density) are difficult to simulate and an altogether compressible framework is needed. Therefore, the compressible Navier-Stokes equations are solved without any additional assumptions. The fully implicit, high-order in space and time, reconstructed discontinuous Galerkin method as implemented within the multi-physics code called ALE3D (Arbitrary Lagrangian and Eulerian in 2D and 3D), developed at Lawrence Livermore National Laboratory (LLNL), is used. This fully implicit, L-stable method accurately captures the compressible nature of the ow in the limit of very low Mach number. It has been widely accepted that above the critical point, only one phase is observed. However, recent research has indicated the existence of the distinct gas-like and liquid-like regions separated by the Widom line, the locus of the maxima of the specific heat. Along the Widom line, density decreases 6-fold, viscosity drops by a factor of 2, while specific heat spikes by an order of magnitude. These variations, specifically in density and viscosity, produce a thick pseudo-interface and flow dynamics behavior akin to film boiling. A pseudo-film at the heated wall of the cavity and the horizontal channel is observed where buoyancy forces induce mixing through the specific configurations. Further the local Rayleigh and Richardson numbers provide maps of the flow field and the buoyancy forces driving the microscopic mixing. In the first chapter, I describe a background of supercritical fluid and the various applications. The second chapter focuses on the mathematical model and numerical method used for simulations, where a description of the equation of state for supercritical water is described. The third chapter focuses on the natural convection cavity with a heated bottom wall. In this cavity a gas-like and a liquid-like flow within the supercritical thermodynamic regime are observed. The fourth chapter focuses on a forced convection, horizontal channel, distinguishing between the gas-like, liquid-like, and mixed flow regimes. Mixed convection flow, with the addition of gravitational forces in the horizontal channel show the influence of variable properties on the hydrodynamic development, heat transfer, and rising instabilities. The last chapter of this research focuses on characterizing the unstable hydrodynamics through time-averaging processes and analysis of the movement of energy through the developing plumes.
We report conventional particle-in-cell (PIC) methods suffer from enhanced numerical heating (explicit PIC) or cooling (semi-implicit PIC) when coupled with a binary Monte-Carlo algorithm for Coulomb collisions. In this work, a fully-implicit θ-PIC scheme (with adjustable time-biasing parameter 1/2 ≤ θ ≤) is considered. The discrete change in energy of a closed system after a time step for this scheme scales with (1/2 - θ)C θ , where C θ is a positive definite quantity that depends on the frequency spectrum of the energy in the fields. Collisions lead to additional energy in the field fluctuations associated with high-frequency light waves produced by a numerical Bremsstrahlung process, which can result in a large increase in the numerical cooling rate for θ > 1/2. However, for θ = 1/2, energy is exactly conserved. The energy in the field fluctuations on long time scales agrees with that calculated using the equipartition theorem for a classical system in thermodynamic equilibrium.
Consideration of the task of computing the invariant subspaces of a given matrix. For this purpose the LU, QR, treppen and bi-iterations have been presented, used, and studied more or less independently of the old-fashioned power method. Each of these methods generates implicitly a sequence of subspaces which determines the convergence properties of the method. The iterations differ in the way in which a basis is constructed to represent each subspace. This aspect largely determines the usefulness of the method. It is shown that the first four iterations produce exactly the same sequence of subspaces as do direct and inverse iteration started from appropriate subspaces. Their convergence properties are therefore the same, and a complete geometric convergence theory is presented in terms of the power method. It is shown that Hessenberg matrices are associated with ideal starting spaces.
A partition procedure for forced-convection conduction transient problems is presented. Mixed time partitions are defined wherein coupled conduction force-matrix equations are discretized using an implicit integration method, followed by derivation of a mixed time integration technique. Explicit-implicit and explicit-explicit partitions are performed for a stability analysis for transient conditions, e.g., those found in an actively air-cooled engine and airframe structure.