Linearized Implicit Scheme Research Articles

Linearized implicit time advancing and defect correction applied to sediment transport simulations

The numerical simulation of sediment transport problems is considered in this paper. The physical problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed profile. The spatial discretization of the governing equations is carried out by a finite-volume method and a modified Roe scheme designed for non-conservative systems. As for the time advancing, starting from an explicit method, a linearized implicit scheme is generated, in which the flux Jacobian is computed through automatic differentiation. Second-order accuracy in space is then obtained through MUSCL reconstruction and in time through a backward differentiation formula associated with a defect-correction approach. The implicit time advancing is compared in terms of accuracy and computational time with the explicit approach for one-dimensional and two-dimensional sediment transport problems, characterized by different time scales for the evolution of the bed and of the water flow. It is shown that, whenever the use of large time steps is compatible with the capture of the water flow dynamics and of the bedload evolution, the implicit scheme is far more efficient than its explicit counterpart with a CPU reduction up to more than two orders of magnitude. This makes implicit time differencing an attractive option for complex real life applications in this area.

Nonlinear scheme with high accuracy for nonlinear coupled parabolic–hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic–hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.

Balanced Milstein Methods for Ordinary SDEs

Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linear-implicit schemes which generate mean square converging numerical approximations with qualitative improvements and global rate 1. 0 of mean square convergence, compared to commonly known numerical methods for SDEs with Lipschitzian coefficients.

Exact factorization technique for numerical simulations of incompressible Navier–Stokes flows

Splitting techniques break an ill-conditioned indefinite system resulting from incompressible Navier–Stokes equations into well-conditioned subsystems, which can be solved reliably and efficiently. Apart from the ambiguity regarding numerical boundary conditions for the pressure (and for intermediate velocities, whenever introduced), splitting techniques usually incur splitting errors which reduce time accuracy. The discrete approach of approximate factorization techniques eliminates the need of numerical boundary conditions and restores time accuracy by an approximate inversion of some matrix in the case of semi-implicit time schemes. For linear implicit, non-linear implicit, and higher-order semi-implicit time schemes, however, approximate factorization techniques are laborious. In this paper, we systematically present a new and straightforward exact factorization technique. The main contributions of this work include: (1) the idea of removing the splitting error or the idea of restoring time accuracy for fully discrete systems, (2) the introduction of the pressure-update type and the pressure-correction type of exact factorization techniques for any time schemes, and (3) an analysis of several established techniques and their relations to the exact factorization technique. The exact factorization technique is implemented with a standard second-order finite volume method and is verified numerically.

Balanced Milstein Methods for Ordinary SDEs

Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linear-implicit schemes which generate mean square converging numerical approximations with qualitative improvements and global rate 1. 0 of mean square convergence, compared to commonly known numerical methods for SDEs with Lipschitzian coefficients.

A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations

AbstractAn efficient numerical method is developed for the numerical solution of non‐linear wave equations typified by the regularized long wave equation (RLW) and its generalization (GRLW). The method developed uses a pseudo‐spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. =10pt An important advantage to be gained from the use of this method, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method is second order in time and all‐order in space. The method presented here is for the RLW equation and its generalized form, but it can be implemented to a broad class of non‐linear long wave equations (Equation (2)), with obvious changes in the various formulae. Test problems, including the simulation of a single soliton and interaction of solitary waves, are used to validate the method, which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. Copyright © 2003 John Wiley & Sons, Ltd.

On the Computation of Compressible Turbulent Flows on Unstructured Grids

An accurate, fast, matrix-free implicit method has been developed to solve compressible turbulent How problems using the Spalart and Allmaras one equation turbulence model on unstructured meshes. The mean-flow and turbulence-model equations are decoupled in the time integration in order to facilitate the incorporation of different turbulence models and reduce memory requirements. Both mean flow and turbulent equations are integrated in time using a linearized implicit scheme. A recently developed, fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The spatial discretization is carried out using a hybrid finite volume and finite element method, where the finite volume approximation based on a containment dual control volume rather than the more popular median-dual control volume is used to discretize the inviscid fluxes, and the finite element approximation is used to evaluate the viscous flux terms. The developed method is used to compute a variety of turbulent flow problems in both 2D and 3D. The results obtained are in good agreement with theoretical and experimental data and indicate that the present method provides an accurate, fast, and robust algorithm for computing compressible turbulent flows on unstructured meshes.

DISCRETE VELOCITY MODEL AND IMPLICIT SCHEME FOR THE BGK EQUATION OF RAREFIED GAS DYNAMICS

We present a numerical method for computing transitional flows as described by the BGK equation of gas kinetic theory. Using the minimum entropy principle to define a discrete equilibrium function, a discrete velocity model of this equation is proposed. This model, like the continuous one, ensures positivity of solutions, conservation of moments, and dissipation of entropy. The discrete velocity model is then discretized in space and time by an explicit finite volume scheme which is proved to satisfy the previous properties. A linearized implicit scheme is then derived to efficiently compute steady-states; this method is then verified with several test cases.

Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries

We present new numerical models for computing transitional or rarefied gas flows as described by the Boltzmann-BGK and BGK-ES equations. We first propose a new discrete-velocity model, based on the entropy minimization principle. This model satisfies the conservation laws and the entropy dissipation. Moreover, the problem of conservation and entropy for axisymmetric flows is investigated. We find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator. Then we propose some models that satisfy these constraints. Owing to these properties, we obtain numerical schemes that are economic, in terms of discretization, and robust. In particular, we develop a linearized implicit scheme for computing stationary solutions of the discrete-velocity BGK and BGK-ES models. This scheme is the basis of a code which can compute high altitude hypersonic flows, in 2D plane and axisymmetric geometries. Our results are analyzed and compared to other methods.

A Fast, Matrix-free Implicit Method for Computing Low Mach Number Flows on Unstructured Grids

A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and Navier-Stokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES + LU−SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both in viscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU−SGS method. The numerical results obtained indicate that the use of the GMRES + LU−SGS method leads to a significant increase in performance over the LU−SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.

Computational methods for some non-linear wave equations

Computational methods based on a linearized implicit scheme and a predictor-corrector method are proposed for the solution of the Kadomtsev–Petviashvili (KP) equation and its generalized from (GKP). The methods developed for the KP equation are applied with minor modifications to the generalized case. An inportant advantage to be gained from the use of the linearized implicit method over the predictor-corrector method which is conditionally stable, is the ability to vary the mesh length, and thereby reducing the computational time. The methods are analysed with respect to stability criteria. Numerical results portraying a single line-soliton solution and the interaction of two-line solitons are reported for the KP equation. Moreover, a lump-like soliton (a solitary wave which decays to zero in all space dimensions) and the interaction of two lump solitons are reported for the KP equation.

A linearized implicit pseudo-spectral method for certain non-linear water wave equations

An efficient numerical method is developed for the numerical solution of non-linear wave equations typified by the third- and fifth-order Korteweg–de Vries equations and their generalizations. The method developed uses a pseudo-spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. An important advantage to be gained from the use of this method over the pseudo-spectral scheme proposed by Fornberg and Whitham (a Fourier transform treatment of the space variable together with a leap-frog scheme in time) which is conditionally stable, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method presented here is for the Korteweg–de Vries equations and their generalized forms, but it can be implemented to a broad class of non-linear wave equations (equation (1)), with obvious changes in the various formulae. To illustrate the application of this method, numerical results portraying a single soliton solution and the collision of two solitons are reported for the third- and fifth-order Korteweg–de Vries equations. © 1998 John Wiley & Sons, Ltd.

Implicit Calculations of an Aeroelasticity Problem

The purpose of this work is to show that a linearized implicit scheme for the flow resolution can be an efficient and accurate method for solving fluid-structure interaction. The fluid is modeled by the Euler equations in two dimensions and the structure by a one (free piston) or a two (NACA0012 airfoil) degrees of freedom system. The schemes are developed using a finite volume/finite element formulation and, stating the moving boundary problem in the space-time domain, the Riemann solver is generalized in a suitable manner. Assuming a modal decomposition for the structure's response, an analytical solution to the equation of motion is obtained. The effects of the linearized implicit scheme on the aeroelastic response are demonstrated on the free piston and the NACA 0012 airfoil problems. In the latter case, we focus on the capability of the linearized implicit scheme to accurately predict the stability limit of the coupled response (wing flutter analysis). Although the above analysis is performed using a rigid transformation, a robust moving mesh strategy is presented for more general 2-D and 3-D deformations.

Fully Coupled Unsteady Mobile Boundary Flow Model

A fully coupled, one‐dimensional river model, capable of predicting sediment‐transport and bed‐level changes under unsteady flow conditions is described. The governing equations for mass and momentum balance of sediment‐water mixture under unsteady flow conditions in natural rivers with irregular geometries, are solved simultaneously using the Preissmann four‐point linear implicit scheme with weighting factors for space and time coordinates. The resulting system of algebraic equations is solved using a solution procedure that bridges upstream and downstream boundary nodal values by successive substitutions. The friction slope is expressed in a consistent form that allows the consideration of the changing alluvial roughness effect. The system of governing equations is explicitly coupled by introducing the term representing the rate of change of bed level. The fully coupled unsteady mobile boundary flow model (FCM) is applied to the classical problem of sediment deposition upstream of a dam with an arbitrar...

On the accuracy of block implicit and operator-splitting algorithms in confined flame propagation problems

The propagation of a one-dimensional, confined, laminar flame is studied by means of two operator-splitting methods, four linear block implicit schemes and the standard implicit and Crank-Nicolson techniques. It is shown that the implicit method underestimates the flame speed and pressure because of first-order temporal truncation errors. Second-order accurate, in both space and time, methods yield nearly the same flame location and pressure as the operator-splitting techniques and the Crank-Nicolson scheme. Block linear methods which are first-order accurate in time and second-order accurate in space show temperature oscillations in the burned gas region near the flame front. These methods are found to be unstable when a delta formulation is used and the flame approaches the combustor wall. The instabilities are attributed to the large magnitude of the source terms and the rate of change of these terms with respect to time. A reduction in the time step brings the results of first-order accurate block met...

A method of parametric correction of difference schemes

The difference discretization of problems of mathematical physics can be performed in different ways, so that manifolds of d.s. arise which may be used for obtaining specific properties such as conservativeness, stability, economy, or a certain order of approximation, etc. We know, however, that A-stability cannot be obtained simultaneously with e.g., the third or higher order of approximation of stiff systems of equations in the class of linear implicit schemes,ormonotonicity simultaneously with second or higher order of approximation for hyperbolic equations in the class of linear d.s. , or stability simultaneously with second or higher order of approximation in the class of explicit schemes for the equations of gas dynamics in Euler variables. We shall consider an approach whereby these difficulties can be to some extent overcome and d.s. with new properties obtained.

On a class of implicit finite-difference schemes

Implicit non-linear monotonic divergence finite-difference approximations are considered for differential equations of first order with respect to time. The schemes are shown to be solvable in the class of bounded functions. Results relevant to the properties of the solutions of linear implicit schemes are obtained.