Truncation Error Estimates Research Articles

Biot consolidation analysis with automatic time stepping and error control Part 1: theory and implementation

SUMMARY A new nite element algorithm for solving elastic and elastoplastic coupled consolidation problems is described. The procedure treats the governing consolidation relations as a system of rst-order di!erential equations and is based on the backward Euler and Thomas and Gladwell schemes with automatic subincrementation of a prescribed series of time increments. The prescribed time increments, which are called coarse time steps, serve to start the procedure and are chosen by the user. The automatic consolidation algorithm attempts to select the time subincrements such that, for a given mesh, the time-stepping (or temporal discretisation) error in the displacements lies close to a specied tolerance. Unlike existing solution techniques, the new algorithm computes not only the displacements and pore pressures, but also their derivatives with respect to time. These extra variables permit a family of unconditionally stable integration algorithms to be constructed which automatically provide an estimate of the local truncation error for each time step. This error estimate is inexpensive to compute and may be used to develop a simple and e$cient automatic time stepping mechanism. For the elastic case, the displacements and pore pressures at the end of each subincrement may be solved directly without the need for iteration. For elastoplastic behaviour, however, the governing relationships are non-linear and a system of non-linear equations must be solved to compute the updates. Copyright ( 1999 John Wiley & Sons, Ltd. The nite element method can be used to model coupled consolidation using a mixed formulation which incorporates displacement and pore pressure variables. Even for elastic material behaviour, the resulting governing equations may be non-linear due to the dependence of the consolidation coe$cient on the excess pore water pressures. The solution of these equations requires the discretization of the time domain into a number of time increments which can be di$cult to

A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms

Higher-order Runge-Kutta (RK) algorithms employing local truncation error (LTE) estimates have had very limited success in solving stiff differential equations. These LTEs do not recognize stiffness until the region of instability has been crossed after which no correction is possible. A new technique has been designed, using the local stiffness function (LSF), which can detect stiffness very early before instability occurs. The LSF is a normalized dimensionless ratio which is essentially based on the product of the step size and the geometric mean of all the slopes. It is exceedingly sensitive to the onset of stiffness. Together, the LSF and the LTE form a complementary pair which can cooperate to help solve some mildly stiff equations which were previously intractable to RK algorithms alone. Examples are given of implementation and LSF performance.

Variable step Rosenbrock algorithm for transient response of damped structures

A Rosenbrock algorithm with varying time step is adapted for transient analysis of damped second-order differential equations. The time step adjustment is based on an embedded local truncation error estimation formula. An interpolation formula can be used for intermediate output. The stepping formula is L-stable and the error estimation formula is bounded for large time steps. The Rosenbrock algorithm is compared with the Thomas—Gladwell STEP34 algorithm, which is found to be only conditionally stable. Numerical results are given for two linear examples: a stiff, linear, two-degree-of-freedom system and a non-proportionally damped plate.

Application of a Hybrid Method to Jet Impinging Phenomena with a Cross Flow. 1st Report. Effects of a Local Refinement Method with a Truncation Error Estimation.

The local refinement technique is very useful method to get high-precision solution with lower efforts in the structural grid system and has been applied to many cases. In using this method, some methods are needed to specify the regions on which the mesh should be fined on the calculation domain and a method based on the truncation error estimation theory was proposed. This method makes it possible to determine the regions with only one, non-iterative calculation and is useful for implicit method. A local mesh refinement method with SIMPLE algorithm was applied to the abovementioned region. This method basically satisfies the conservation law on the fine/coarse cell boundaries. As a result, high-precision solution was got with shorter time and smaller memory in comparison with the usual method.

Precision corrections to dispersive bounds on form factors

We present precision corrections to dispersion relation bounds on form factors in bottom hadron semileptonic decays and analyze their effects on parameterizations derived from these bounds. We incorporate QCD two-loop and nonperturbative corrections to the two-point correlator, consider form factors whose contribution to decay rates is suppressed by lepton mass, and implement more realistic estimates of truncation errors associated with the parameterizations. We include higher resonances in the hadronic sum that, together with heavy quark symmetry relations near zero recoil, further tighten the sum rule bounds. Utilizing all these improvements, we show that each of the six form factors in B --> D l nu and B --> D^* l nu can be described with 3% or smaller precision using only the overall normalization and one unknown parameter. A similar one-coefficient parameterization of one of the Lambda_b --> Lambda_c l nu form factors, together with heavy quark symmetry relations valid to order 1/m^2, describes the differential baryon decay rate in terms of one unknown parameter and the phenomenologically interesting quantity (\bar Lambda)_Lambda \approx M_{Lambda_b} - m_b. We discuss the validity of slope-curvature relations derived by Caprini and Neubert, and present weaker, corrected relations. Finally, we present sample fits of current experimental B --> D^*l nu and B --> D l nu data to the improved one-parameter expansion.

Analysis and implementation of TR-BDF2

Bank et al. (1985) developed a one-step method, TR-BDF2, for the simulation of circuits and semiconductor devices based on the trapezoidal rule and the backward differentiation formula of order 2 that provides some of the important advantages of BDF2 without the disadvantages of a memory. Its success and popularity in the context justify its study and further development for general-purpose codes. Here the method is shown to be strongly S-stable. It is shown to be optimal in a class of practical one-step methods. An efficient, globally C 1 interpolation scheme is developed. The truncation error estimate of Bank et al. (1985) is not effective when the problem is very stiff. Coming to an understanding of this leads to a way of correcting the estimate and to a more effective implementation. These developments improve greatly the effectiveness of the method for very stiff problems.

Stability and error analysis for time integrators applied to strain-softening materials

The behavior of explicit time integrators is examined for the field equations of dynamic viscoplasticity in one dimension. The analysis proceeds by linearizing the field equations about the current state and freezing coefficients so that the solution to the system of ordinary differential equations can be computed explicitly. Amplification matrices for the central difference method (with rate tangent constitutive update) and fourth-order Runge-Kutta schemes are obtained by applying these schemes to the resulting linear, constant coefficient equations. These amplification matrices lead to local truncation error estimates for the temporal integrators as well as to estimates of the resulting growth rate. Analysis of the cental difference method with a forward Euler constitutive update indicates that the local truncation error involves as the product of the third power of the strain-rate and the square of the time step. In viscoplastic models, the viscoplastic strain-rate is an increasing (decreasing) function of the viscoplastic strain in softening (hardening). Also, the eigenvalues of the rate tangent method increase with strain-rate in softening. Thus, it is recommended that the time step be reduced in localization problems to keep the strain-increment within a prescribed tolerance. As a byproduct of the analysis, conditions for numerical stability for the central difference method with a rate tangent constitutive update are obtained for strain-hardening dynamic viscoplasticity. In the softening regime, the resulting linearized differential equations admit exponentially growing solution. Thus, conventional definitions of stability such as Neumann stability and absolute stability are inappropriate. In order to characterize stability of such systems, we introduce the concept of g-rel stability which combines the concept of relative and absolute stability. We show conditions under which the resulting system is g-rel stable and thus characterize the nature of the numerical stability that can be expected in such problems.

Solution-adaptive techniques in computational heat transfer and fluid flow

Recent contributions to solution-adaptive grid and solution-adaptive differencing techniques are breifly described in this paper. In solution-adaptive grid techniques, the grid points are dynamically reclustered or refined to improve the resolution in the important regions where the truncation error estimate is high. In solution-adaptive differencing techniques, the order of the differencing scheme in high error estimate region is dynamically increased. Thus both adaptive grid and adaptive differencing techniques represent error-equidistribution procedures. Two strategies for adaptive gridding are described in this paper. In one strategy, termed Global Adaptation, all grid points participate in the grid point redistribution process. In the other strategy, called Local Adaptation, grid refinement is performed only in local regions with high truncation error estimates. Results of various problems are presented which show the improvements obtained with solution-adaptive techniques.

On an Adaptive Time Stepping Strategy for Solving Nonlinear Diffusion Equations

A new time step selection procedure is proposed for solving nonlinear diffusion equations. It has been implemented in the ASWR finite element code of Lorenz and Svoboda [10] for 2D semiconductor process modelling diffusion equations. The strategy is based on equidistributing the local truncation errors of the numerical scheme. The use of B-splines for interpolation (as well as for the trial space) results in a banded and diagonally dominant matrix. The approximate inverse of such a matrix can be provided to a high degree of accuracy by another handed matrix, which in turn can be used to work out the approximate finite difference scheme corresponding to the ASWR finite element method, and further to calculate estimates of the local truncation errors of the numerical scheme. Numerical experiments on six full simulation problems arising in semiconductor process modelling have been carried out. Results show that our proposed strategy is more efficient and better conserves the total mass.

Adaptive Grid Refinement for Two-Dimensional and Three-Dimensional Nonhydrostatic Atmospheric Flow

Although atmospheric phenomena tend to be localized in both time and space, numerical models generally employ only uniform discretizations or fixed nested grids. An adaptive grid technique implemented in 2D and 3D nonhydrostatic elastic atmospheric models is described. The adaptive technique makes use of separate rectangular refinements to increase resolution where truncation error estimates are large. Multiple, rotated, overlapping grids are used along with an arbitrary number of discrete grid-refinement levels. Refinements are placed and removed automatically during the integration based an estimates of the truncation error in the evolving solution. The technique can be viewed as an extension of the nesting technique often used in atmospheric models. The adaptive model integrates the compressible, nonhydrostatic equations of motion. Although sound waves are not significant in the solution, they do constrain the time step. A splitting technique is used to accommodate the sound waves by advancing certain terms with a separate smaller time step. The terms responsible for gravity waves are also integrated with the smaller time step, and with the acoustic modes filtered through the use of divergence damping, the resulting model can be run as efficiently as hydrostatic models. Boundary conditions developed for the splitting technique in the adaptive framework are described and tested in the 2D and 3D models. The adaptive technique is shown to be efficient when compared to single fixed-grid simulations. Two new features are included in the basic solver. Also considered are additional complications that arise because of the necessary use of parameterized physics. The dependence of many parameterizations on grid scale creates difficulties in evaluating truncation error and raises more general questions concerning solution error in nested and adaptive models.

Segmented multigrid domain decomposition procedure for incompressible viscous flows

AbstractThe application of grid stretching or grid adaptation is generally required in order to optimize the distribution of nodal points for fluid‐dynamic simulation. This is necessitated by the presence of disjoint high gradient zones, that represent boundary or free shear layers, reversed flow or vortical flow regions, triple deck structures, etc. A domain decomposition method can be used in conjunction with an adaptive multigrid algorithm to provide an effective methodology for the development of optimal grids. In the present study, the Navier‐Stokes (NS) equations are approximated with a reduced Navier‐Stokes (RNS) system, that represents the lowest‐order terms in an asymptotic Re expansion. This system allows for simplified boundary conditions, more generality in the location of the outflow boundary, and ensures mass conservation in all subdomain grid interfaces, as well as at the outflow boundary. The higher‐order (NS) diffusion terms are included through a deferred corrector, in selected subdomains, when necessary. Adaptivity in the direction of refinement is achieved by grid splitting or domain decomposition in each level of the multigrid procedure. Normalized truncation error estimates of key derivatives are used to determine the boundaries of these subdomains. The refinement is optimized in two co‐ordinate directions independently. Multidirectional adaptivity eliminates the need for grid stretching so that uniform grids are specified in each subdomain. The overall grid consists of multiple domains with different meshes and is, therefore, heavily graded. Results and computational efficiency are discussed for the laminar flow over a finite length plate and for the laminar internal flow in a backward‐facing step channel.

A formula for estimation of truncation errors of convection terms in a curvilinear coordinate system

A formula for truncation errors of convection terms arising from the Navier-Stokes equation in a curvilinear coordinate system is derived. These truncation error terms are interpreted in terms of grid size, grid uniformity, grid line angle, flow angle, and derivatives of flow properties to gain physical insights from the formula. The role of these factors in determining truncation errors is discussed. It is shown that an optimal grid arrangement cannot be obtained without considering the interactions between the grid and the flow field. The effect of the grid orthogonality on truncation errors is also analyzed for simple cases. The derived formula provides a useful indicator for truncation error distribution which yields guidelines for grid adaption.

Multigrid methods and adaptive refinement techniques in elliptic problems by finite element methods

Multigrid and adaptive refinement techniques are powerful tools in the resolution of problems arising from computational mechanics. In structured grids the number of numerical operations needed in the resolution via the multigrid method is of order N, the number of degrees of freedom, regardless the dimensionality of the problem. In this work a multigrid technique, which works on unstructured meshes, is described. These meshes are obtained from a user defined tesselation through bisection refinement upon any element. Original techniques for nodes numbering, irregular nodes detection and nodes classification are developed in view of multigrid method implementation. The error indicator used in the refinement process is based on truncation error estimates. The refinement strategy is oriented toward error reduction at constant computational effort. Finally several numerical examples are presented.

Recent progress in estimating uncertainty in geomagnetic field modeling

Abstract Recent work pertaining to estimating error and accuracies in geomagnetic field modeling is reviewed from a unified viewpoint and illustrated with examples. The formulation of a finite dimensional approximation to the underlying infinite dimensional problem is developed. Central to the formulation is an inner product and norm in the solution space through which a priori information can be brought to bear on the problem. Such information is crucial to estimation of the effects of higher degree fields at the Core-Mantle boundary (CMB) because the behavior of higher degree fields is masked in our measurements by the presence of the field from the Earth's crust. Contributions to the errors in predicting geophysical quantities based on the approximate model are separated into three categories: (1) the usual error from the measurement noise; (2) the error from unmodeled fields, i.e. from sources in the crust, ionosphere, etc.; and (3) the error from truncating to a finite dimensioned solution and prediction space. The combination of the first two is termed low degree error while the third is referred to as truncation error. The error analysis problem consists of “characterizing” the difference δz = z—z, where z is some quantity depending on the magnetic field and z is the estimate of z resulting from our model. Two approaches are discussed. The method of Confidence Set Inference (CSI) seeks to find an upper bound for |z—ž|. Statistical methods, i.e. Bayesian or Stochastic Estimation, seek to estimate E(δz2 ), where E is the expectation value. Estimation of both the truncation error and low degree error is discussed for both approaches. Expressions are found for an upper bound for |δz| and for E(δz2 ). Of particular interest is the computation of the radial field, B., at the CMB for which error estimates are made as examples of the methods. Estimated accuracies of the Gauss coefficients are given for the various methods. In general, the lowest error estimates result when the greatest amount of a priori information is available and, indeed, the estimates for truncation error are completely dependent upon the nature of the a priori information assumed. For the most conservative approach, the error in computing point values of Br at the CMB is unbounded and one must be content with, e.g., averages over some large area. The various assumptions about a priori information are reviewed. Work is needed to extend and develop this information. In particular, information regarding the truncated fields is needed to determine if the pessimistic bounds presently available are realistic or if there is a real physical basis for lower error estimates. Characterization of crustal fields for degree greater than 50 is needed as is more rigorous characterization of the external fields.

Implicit numerical integration algorithms with de Casteljau predictors

AbstractA common approach for solving initial value problems representing algebraic differential equations is to formulate the solution as a polynomial function. This paper presents predictor based integration formulae by formulating the solution function as a vector valued Bernstein polynomial, namely a Bézier function. The Bézier function subdivision theorem leads to a predictor evaluation algorithm for the next solution point using de Casteljau's iterative control vertex generation technique. It is shown that the convergence of the algorithms is a consequence of the use of the Bernstein uniform approximation basis and that the de Casteljau predictor evaluation scheme leads to a computationally simple estimate of the local truncation error.

A formula for estimation of truncation errors of convection terms in a curvilinear coordinate system

A formula for estimation of truncation errors of convection terms in a curvilinear coordinate system

Algorithms for the rapid computation of response of an oscillator with bounded truncation error estimates

In this paper an algorithm is developed for the accurate and efficient computation of the response of a linear SDOF, subjected to an arbitrary excitation, known deterministically as a set of discrete ordinates sampled uniformly in time. Bounded local truncation error estimates have been established, and indicate that the algorithm has the same order of relative error as currently published methods, whilst requiring far fewer multiplications per time step. Furthermore, two recursive algorithms have been developed for the rapid computation of the response spectrum defined uniquely by the given exciting time history. Both algorithms provide accurate estimates of response ordinates in sensitive regions of the spectra, while avoiding the calculation of large numbers of spectral values in areas showing only minor irregularities. Estimates of error bounds have been derived for each method. Finally, results from the algorithms presented in this paper have been compared to closed solutions for SDOF having homogeneous initial conditions. The results indicate a good agreement within the expected tolerance.

Short-rise-time microwave pulse propagation through dispersive biological media

Numerical research is reported on the propagation of short microwave pulses into living, biological materials. These materials are dispersive, and data on the dielectric constant and conductivity for these materials follow a Debye model. A Fourier-series calculation is presented that predicts the occurrence of Brillouin precursors when the incident pulses have sufficiently short rise times. These transients are attenuated with increasing propagation distance but are attenuated more slowly than the carrier frequency of the pulse, which is attenuated exponentially with distance. An analysis of the numerical error resulting from truncation of the Fourier series is given. Upper-bound estimates of truncation error show good series convergence.

Truncation Error Estimates for Refinement Criteria in Nested and Adaptive Models

Abstract Truncation error estimates are considered as criteria for fine-grid placement and movement in nested and adaptive finite-difference atmospheric models. A simple method for calculating the truncation error at any time during an integration is described. Two cases using the shallow-water equations and the hydrostatic primitive equations are examined to demonstrate the accuracy of the method and illuminate the relationships among the truncation error, a particular discretization, the equations being solved and the flow physics. The relationship between the truncation error and the solution error is also discussed and it is argued that minimization of the truncation error is the necessary consideration for producing more accurate numerical solutions. Examples of use of the truncation error estimates in adaptive models are also presented.

Adaptive grid refinement for numerical weather prediction

An adaptive atmospheric flow model is described and results of integrations with this model are presented. The adaptive technique employed is that of Berger and Oliger. The technique uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids which have been placed based on a Richardson-type estimate of the truncation error in the coarse grid solution. By correctly coupling the integrations on the various grids, periodically re-estimating the error, and recreating the finer grids, uniformily accurate solutions are economically produced. The “primitive” hydrostatic equations of meteorology are solved for the advection of a barotropic cyclone and for the development of a baroclinic disturbance which results from the perturbation of an unstable jet. These integrations demonstrate the feasibility of using multiple, rotated, overlapping fine grids. Direct computations of the truncation error are used to confirm the accuracy of the Richardson-type truncation error estimates.