Von Neumann Analysis Research Articles

A Study of the von Neumann Stability Analysis of a Semi-implicit Numerical Method Applied Over the Radial Impurity Transport Equation in Tokamak Plasma

The radial impurity transport equation for tokamak plasma is a form of diffusion–convection–reaction equation. The impurity transport equation is solved to determine the distribution of impurity (non-fuel) ion species with different ionization states perpendicular to magnetic surfaces of tokamak plasma. The equation for each charge (ionization) state Z is a non-linear, second-order in space, first-order in time, parabolic partial differential equation coupled to the previous Z − 1 and the next Z + 1 charge states of the impurity species through its reaction term. The number of differential equations to be solved simultaneously is hence determined by the number of ionization states of the impurity species studied. The solution to the set of these coupled equations can be obtained using a semi-implicit numerical method applied on it. The present study describes the application of von Neumann stability analysis over the semi-implicit numerical method applied over the radial impurity transport equation and determines a generic stability criterion for the method. The stability analysis is further illustrated using the geometry of Aditya tokamak installed at the Institute for Plasma Research Gandhinagar, India as an example. The impurity species considered is oxygen (Atomic number = 8). This leads to a set of eight coupled equations for charge states Z = 1 to 8 over which von Neumann analysis is illustrated in present study.

Numerical solutions of the Gardner equation by extended form of the cubic B-splines

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. The optimum value of the extension parameter can be determined by scanning some intervals containing zero. This study aims to solve some initial boundary value problems constructed for the Gardner equation by the extended cubic B-spline collocation method. The test problems are derived from some analytical studies to validate the efficiency and accuracy of the suggested method. The conservation laws are also determined to observe whether the test problems remain constant as expected from the theoretical aspect. The stability of the proposed method is investigated by the von Neumann analysis.

Comparison of artificial-dissipation and solution-filtering stabilization schemes for time-accurate simulations

The current study investigates the use of solution filtering and artificial dissipation as stabilization methods for time-accurate computations, notably focusing on how the decoupling of dissipation from integration impacts simulation error. Rewriting solution filtering in an effective artificial-dissipation form explicitly reveals the issue of temporal inconsistency, which is addressed by a proposed CFL re-scaling. In addition, expressing symmetric discrete filters as difference operators inspires the derivation of a “filter-based” artificial-dissipation formulation that provides the opportunity for direct spectral manipulation by utilizing known filter stencil specifications; this furthermore facilitates the creation of Padé-type artificial dissipation terms with spectrally tunable and scale-discriminant properties. Damping characteristics of the schemes are assessed through von Neumann analysis and are confirmed via simulations of the one-dimensional advection equation (wave-packet transport) as well as the three-dimensional compressible Navier–Stokes system (Taylor–Green vortex), where the impact of time step size on accuracy of the respective stabilization techniques is inspected.

Modified cubic B-spline quasi-interpolation numerical scheme for hyperbolic conservation laws

ABSTRACTIn general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified version of the cubic B-spline quasi-interpolation (CBSQI) numerical scheme for the hyperbolic conservation laws in one space dimension. In order to stabilize the CBSQI scheme, we have added an adaptive artificial viscosity term and derived the conditions under which the modified scheme is monotone. Although the CBSQI scheme can be of at most fourth order accurate in space, the monotone condition restricts the resulting scheme to be of at most order of one, as expected. In order to improve the order of accuracy, we adopt a procedure, which we call the weighted adaptive viscosity approach, through which we achieved the fourth order accuracy both in space and time in the modified CBSQI scheme. Further, linear -stability of the modified CBSQI scheme is analyzed using von-Neumann analysis. Numerical experiments are performed to validate the efficiency and stability of the modified CBSQI scheme.

Solutions of the fractional combined KdV\u2013mKdV equation with collocation method using radial basis function and their geometrical obstructions

The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV–mKdV) equation is obtained by using the (1/G^{prime}) expansion method. To investigate a geometrical surface of the exact solution, we choose gamma=1. The collocation method is applied to the fractional combined KdV–mKdV equation with the help of radial basis for 0<gamma<1. L_{2} and L_{infty} error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz–Minkowski space.

An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

This paper displays the approach of the time-splitting Fourier spectral (TSFS) technique for the linear Riesz fractional Schrodinger equation (RFSE) in the semi-classical regime. The splitting technique is shown to be unconditionally stable. Further a suitable implicit finite difference discretization of second order has been manifested for the RFSE where the Riesz derivative has been discretized via an approach of fractional centered difference. Moreover the stability analysis for the implicit scheme has also been presented here via von Neumann analysis. The L2-norm and $L^{\infty}$ -norm errors are calculated for $\vert u(x,t)\vert^{2}$ , Re $(u(x,t))$ and Im $(u(x,t))$ for various cases. The results obtained by the methods are further tabulated for the absolute errors for $\vert u(x,t)\vert^{2}$ . Furthermore the graphs are depicted showing comparison of $\vert u(x,t)\vert^{2}$ by both techniques. The derivatives are taken here in the context of the Riesz fractional sense. Apart from that, the comparative study put forth in the following section via tables and graphs between the implicit second-order finite difference method (IFDM) and the TSFS method is for the purpose of investigating the efficiency of the results obtained. Moreover the stability analysis of the presented techniques manifesting their unconditional stability makes the proposed approach more competing and accurate.

Engineering a lunar photolithoautotroph to thrive on the moon – life or simulacrum?

AbstractRecent work in developing self-replicating machines has approached the problem as an engineering problem, using engineering materials and methods to implement an engineering analogue of a hitherto uniquely biological function. The question is – can anything be learned that might be relevant to an astrobiological context in which the problem is to determine the general form of biology independent of the Earth. Compared with other non-terrestrial biology disciplines, engineered life is more demanding. Engineering a self-replicating machine tackles real environments unlike artificial life which avoids the problem of physical instantiation altogether by examining software models. Engineering a self-replicating machine is also more demanding than synthetic biology as no library of functional components exists. Everything must be constructedde novo. Biological systems already have the capacity to self-replicate but no engineered machine has yet been constructed with the same ability – this is our primary goal. On the basis of the von Neumann analysis of self-replication, self-replication is a by-product of universal construction capability – a universal constructor is a machine that can construct anything (in a functional sense) given the appropriate instructions (DNA/RNA), energy (ATP) and materials (food). In the biological cell, the universal construction mechanism is the ribosome. The ribosome is a biological assembly line for constructing proteins while DNA constitutes a design specification. For a photoautotroph, the energy source is ambient and the food is inorganic. We submit that engineering a self-replicating machine opens up new areas of astrobiology to be explored in the limits of life.

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both (n)th and (n+1)th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.

On the convergence of iterative solvers for polygonal discontinuous Galerkin discretizations

We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with that of traditional triangular elements. We solve the semi-discrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi's method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.

Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel

In this paper, an approximate solution for solving weakly singular kernel partial integro-differential equations with time fractional order is proposed. The method is based on using a second-order time difference approximation followed by applying the fractional integral operator and piecewise linear interpolation to compute the singularity of the kernel that appear in the discretization process. The stability of the method is also considered in the sense of von Neumann analysis. Numerical examples are solved to demonstrate the validity and applicability of the presented technique.

Compact Direct Flux Reconstruction for Conservation Laws

In this study, a high-order discontinuous compact direct flux reconstruction (CDFR) method is developed to solve conservation laws numerically on unstructured meshes. Without explicitly constructing any polynomials, the CDFR method directly calculates the nodal flux derivatives via the compact finite difference approach within elements. To achieve an efficient implementation, the construction procedure of flux derivatives is conducted in a standard element. As a result, a standard flux-derivative-construction matrix can be formulated. The nodal flux derivatives can be directly constructed through the multiplication of this matrix and the flux vectors. It is observed that the CDFR method is identical with the direct flux reconstruction method and the nodal flux reconstruction–discontinuous Galerkin method if Gauss–Legendre points are selected as solution points for degrees p up to 8 tested. A von Neumann analysis is then performed on the CDFR method to demonstrate its linear stability as well as dissipation and dispersion properties for linear wave propagation. Finally, numerical tests are conducted to verify the performance of the CDFR method on solving both steady and unsteady inviscid flows, including those over curved boundaries.

Long-time instability in the Runge-Kutta algorithm for a Nosé-Hoover heat bath of a harmonic chain and its stabilization.

In this paper, we investigate the Runge-Kutta algorithm for the Nosé-Hoover heat bath of a harmonic chain. The Runge-Kutta algorithm is found to be unstable in long-time calculations, with the system temperature growing exponentially. The growth rate increases if time step size is chosen larger. By analyzing the Fourier spectra in both space (wave number) and time (frequency), we discover that the growth is caused by spurious energy accumulation, particularly at the largest wave number. Such accumulation may be explained by von Neumann analysis for an infinite chain, with the nonlinear heat bath being ignored. Furthermore, we propose to add a filter to remove excessive energy, which effectively stabilizes the algorithm.

On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations (Au=b) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of P relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of M consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method where the matrix of the system of equations is preconditioned multiplying it by D=diag(A). Our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix A (or the corresponding iteration matrix of the underlying weighted Jacobi scheme) is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis of the continuous elliptic operator. This set of weights is also the optimal one for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. The amplification factor of the method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter ω in the Successive Overrelaxation (SOR) method in some cases. Finally, we demonstrate with practical examples that our method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed (e.g., a 9- or 17-points discretization). This is of interest since the former discretizations do not yield consistently ordered A matrices and, hence, the theory of Young cannot be used to predict the optimal value of the SOR parameter. Furthermore, the optimal SRJ schemes deduced here are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation.

On the behaviour of fully-discrete flux reconstruction schemes

In this study we employ von Neumann analyses to investigate the dispersion, dissipation, group velocity, and error properties of several fully-discrete flux reconstruction (FR) schemes. We consider three FR schemes paired with two explicit Runge–Kutta (ERK) schemes and two singly diagonally implicit RK (SDIRK) schemes. Key insights include the dependence of high-wavenumber numerical dissipation, relied upon for implicit large eddy simulation (ILES), on the choice of temporal scheme and time-step size. Also, the wavespeed characteristics of fully-discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time-step size are investigated. Salient properties from the aforementioned theoretical analysis are then demonstrated in practice using linear advection test cases. Finally, a Burgers turbulence test case is used to demonstrate the importance of the temporal discretization when using FR schemes for ILES.

Von Neumann stability analysis of the u–p reproducing kernel formulation for saturated porous media

This paper introduces the von Neumann method to investigate the temporal stability of the displacement---pressure $$(u{-}p)$$(u-p) reproducing kernel formulation for saturated porous media. Both dynamic and quasi-static formulations are considered and the critical time steps are derived. The effect of lumped and consistent matrices on temporal stability is analyzed under explicit temporal discretization. It is shown that lumped matrices have better temporal stability than consistent matrices. The study also shows that nodal support size greatly affects the critical time step size of the formulations. For consistent matrices, larger support size results in smaller critical time step size; however, opposite relation occurs if lumped scheme is used. The numerical study shows that stabilization parameter of the stabilized nodal integration methods reduces the critical time step size. Transient analyses are performed to verify the results from von Neumann analysis.

A numerical analysis of the nodal Discontinuous Galerkin scheme via Flux Reconstruction for the advection-diffusion equation

Abstract A von Neumann analysis of the Flux Reconstruction (FR) formulation is performed for the linear advection-diffusion equation to investigate the stability, dissipation and dispersion associated with the nodal Discontinuous Galerkin (DG) scheme. We show that the maximum stable time step for advection-diffusion is stricter than that for pure-advection or pure-diffusion individually. However, the simple harmonic sum of the maximum stable time steps for advection and diffusion provides a suitable estimate for the linear advection-diffusion and Navier–Stokes equations on unstructured, tensor product elements. The estimates are accurate within 50% error on all test cases and are conservative on tests with Cartesian grids but not always on unstructured grids. We show that the CFL limit is strongly influenced by the choice of interface fluxes and, in general, the limit for a scheme using centered values is much higher than that which has one-sided values. We also verify that schemes with centered values produce less error for well resolved solutions while schemes with one-sided values produce less error for solutions that are under-resolved.

Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations

In this paper, we consider an exactly divergence-free scheme to solve the magnetic induction equations. This problem is motivated by the numerical simulations of ideal magnetohydrodynamic (MHD) equations, a nonlinear hyperbolic system with a divergence-free condition on the magnetic field. Computational methods without satisfying such condition may lead to numerical instability. One class of methods, constrained transport schemes, is widely used as divergence-free treatments. So far there is not much analysis available for such schemes. In this work, we take an exactly divergence-free scheme proposed by [Li and Xu, J. Comput. Phys. 231 (2012) 2655–2675] as a candidate of the constrained transport schemes, and adapt it to solve the magnetic induction equations. For the resulting scheme applied to the equations with a constant velocity field, we carry out von Neumann analysis for numerical stability on uniform meshes. We also establish the stability and error estimates based on energy methods. In particular, we identify the stability mechanism due to the spatial and temporal discretizations, and the role of the exactly divergence-free property of the numerical solution for stability. The analysis based on energy methods can be extended to non-uniform meshes, and they can also be applied to the magnetic induction equations with a variable velocity field, which is more relevant to the MHD simulations.

Analysis of velocity-components decoupled projection method for the incompressible Navier–Stokes equations

We study the temporal accuracy and stability of the velocity-components decoupled projection method (VDPM) for fully discrete incompressible Navier–Stokes equations. In particular, we investigate the effect of three formulations of the nonlinear convection term, which include the advective, skew-symmetric, and divergence forms, on the temporal accuracy and stability. Second-order temporal accuracy of the VDPM for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l2-norm. Considering the energy evolution, we demonstrate that the VDPM is stable when the time step is less than or equal to a constant. Stability diagrams, which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices, are obtained using von Neumann analysis. These diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term. Numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability, and the effects of the formulations of the nonlinear convection term are analyzed. Overall, our results indicate that the VDPM along with an advective discrete convection operator is almost unconditionally stable, second-order accurate in time, and computationally efficient because of the non-iterative solution procedure in solving the decoupled momentum equations.

Instability of the finite-difference split-step method applied to the nonlinear Schrödinger equation. I. standing soliton

We consider numerical instability that can be observed in simulations of solitons of the nonlinear Schrödinger equation (NLS) by a split-step method (SSM) where the linear part of the evolution is solved by a finite-difference discretization. The von Neumann analysis predicts that this method is unconditionally stable on the background of a constant-amplitude plane wave. However, simulations show that the method can become unstable on the background of a soliton. We present an analysis explaining this instability. Both this analysis and the features and threshold of the instability are substantially different from those of the Fourier SSM, which computes the linear part of the NLS by a spectral discretization. For example, the modes responsible for the numerical instability are not similar to plane waves, as for the Fourier SSM or, more generally, in the von Neumann analysis. Instead, they are localized at the sides of the soliton. This also makes them different from “physical” (as opposed to numerical) unstable modes of nonlinear waves, which (the modes) are localized around the “core” of a solitary wave. Moreover, the instability threshold for thefinite-difference split-step method is considerably relaxed compared with that for the Fourier split-step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1002–1023, 2016

Instability of the finite-difference split-step method applied to the nonlinear Schrödinger equation. II. moving soliton

We analyze a mechanism and features of a numerical instability (NI) that can be observed in simulations of moving solitons of the nonlinear Schrödinger equation (NLS). This NI is completely different than the one for the standing soliton. We explain how this seeming violation of the Galilean invariance of the NLS is caused by the finite-difference approximation of the spatial derivative. Our theory extends beyond the von Neumann analysis of numerical methods; in fact, it critically relies on the coefficients in the equation for the numerical error being spatially localized. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1024–1040, 2016