Achieving high-order fluctuation splitting schemes by extending the stencil

An extension to the fluctuation splitting approach for approximating hyperbolic conservation laws is described, which achieves higher than second-order accuracy in both space and time by extending the range of the distribution of the fluctuations. Initial results are presented for a simple linear scheme which is third-order accurate in both space and time on uniform triangular grids. Numerically induced oscillations are suppressed by applying the flux-corrected transport algorithm. These schemes are evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent flows and some suggestions for their future development are made.     

The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A three-stage second-order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2, 3. It has advantages in the low accuracy range, and this range is interesting for an application in splitting methods. Numerical results are presented.     

Asymptotic-preserving schemes for unsteady flow simulations

Asymptotic preserving schemes are proposed for numerical simulation of unsteady flows in the relaxation framework. Using a splitting operator to treat the transport and the collision terms separately, we reconstruct a stable and accurate method that converges uniformly to the correct equilibrium. This convergence is also ensured when the relaxation time is unresolved (asymptotic preserving). Numerical results and comparisons are shown for the multidimensional Euler system of gas dynamics.     

Second-order explicit characteristic difference schemes for quasilinear hyperbolic systems

We present two-step, second-order explicit characteristic difference schemes for the numerical solution of initialvalue problems for quasilinear hyperbolic system and show that the method is stable for systems with constant coefficients.     

Towards optimal explicit time-stepping schemes for the gyrokinetic equations

The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes measures to improve the efficiency of such computations, thereby making them more realistic. Explicit Runge–Kutta schemes are considered to be well suited for time-stepping. Although the numerical algorithms are often highly optimized, performance can still be improved by a suitable choice of the time-stepping scheme, based on the spectral analysis of the underlying operator. Here, an operator splitting technique is introduced to combine first-order Runge–Kutta–Chebychev schemes for the collision term with fourth-order schemes for the remaining terms. In the nonlinear regime, based on the observation of eigenvalue shifts due to the (generalized) E×B$E\times B$ advection term, an accurate and robust estimate for the nonlinear timestep is developed. The presented techniques can reduce simulation times by factors of up to three in realistic cases. This substantial speedup encourages the use of similar timestep optimized explicit schemes not only for the gyrokinetic equation, but also for other applications with comparable properties.     

Implicit–explicit numerical schemes for jump–diffusion processes

Abstract We study the numerical approximation of solutions for parabolic integro-differential equations (PIDE). Similar models arise in option pricing, to generalize the Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term, unconditionally stable implicit difference schemes are not practically feasible. Here we propose using implicit-explicit (IMEX) Runge-Kutta methods for the time integration to solve the integral term explicitly, giving higher-order accuracy schemes under weak stability time-step restrictions. Numerical tests are presented to show the computational efficiency of the approximation. Mathematics Subject Classification (1991): Primary: 65M12; Secondary: 35K55, 49L25     

Space-time adaptive simulations for unsteady Navier-Stokes problems

In this paper we apply to the unsteady Navier-Stokes problems some results concerning a posteriori error estimates and adaptive algorithms known for steady Navier-Stokes, unsteady heat and reaction-convection-diffusion equations and unsteady Stokes problems. Our target is to investigate the real viability of a fully combined space and time adaptivity for engineering problems. The comparison between our numerical simulations and the literature results demonstrates the accuracy and efficiency of this adaptive strategy.     

Fast group explicit algorithms for diffusion problems

In this paper fast algorithms based on the Group Explicit (GE) formula are derived and the implementation procedure described for the diffusion problem. The new algorithms have computational cost which is half that of the classical explicit method. Furthermore they are shown to be as accurate as the earlier methods with reduced stability problems.     

Fractional step artificial compressibility schemes for the unsteady incompressible Navier-Stokes equations

We propose a novel, time-accurate approach for solving the unsteady, three-dimensional, incompressible Navier-Stokes equations on non-staggered grids. The approach modifies the standard, dual-time stepping artificial-compressibility (AC) iteration scheme by incorporating ideas from pressure-based, fractional-step (FS) formulations. The resulting hybrid fractional-step/artificial-compressibility (FSAC) method is second-order accurate and advances the Navier-Stokes equations in time via a two-step procedure. In the first step, which is identical to the convection-diffusion step in pressure-based FS methods, a preliminary velocity field is calculated, which is not divergence-free. In the second step, however, instead of deriving a pressure-Poisson equation as in FS methods, the projection of the velocity field into the solenoidal vector space is implemented using a dual-time stepping AC formulation. Unlike the standard dual-time stepping AC formulations, where the dual-time iterations are carried out with the entire non-linear system, in the FSAC scheme the convective and viscous terms are computed only once or twice per physical time step. Numerical experiments show that the proposed method provides second-order accurate solutions and requires considerably less CPU time than the widely used standard AC formulation. To demonstrate its ability to compute complicate problems, the method is also applied to a flow past cylinder with endplates.     

A domain splitting algorithm for parabolic problems

In the parallel implementation of solution methods for parabolic problems one has to find a proper balance between the parallel efficiency of a fully explicit scheme and the need for stability and accuracy which requires some degree of implicitness. As a compromise a domain splitting scheme is proposed which is locally implicit on slightly overlapping subdomains but propagates the corresponding boundary data by a simple explicit process. The analysis of this algorithm shows that it has satisfactory stability and approximation properties and can be effectively parallelized. These theoretical results are confirmed by numerical tests on a transputer system.     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.     

Application of explicit schemes for the simulation of the two-phase filtration process

In the paper, the new model based on the kinetic approach is proposed to describe the process of two-phase slightly compressible fluid filtration. The capillary and gravity forces are taken into account. The obtained hyperbolic continuity equations for phase liquids are approximated by the explicit three-level schemes with a sufficiently mild stability condition. Due to its logical simplicity, the computational algorithm can be easily adapted to the hybrid architecture of modern supercomputers. The results of computations on a graphics accelerator are presented for the problem on contaminant infiltration into water-saturated soil and the high parallelization efficiency is demonstrated.     

Approximation schemes for deal splitting and covering integer programs with multiplicity constraints

We consider the problem of splitting an order for R goods, R≥1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting with price tables (DST), the buyer can generate such combinations using price tables. Our problems, which often occur in online reverse auctions, generalize covering integer programs with multiplicity constraints (CIP), where we must fill up an R-dimensional bin by selecting (with a bounded number of repetitions) from a set of R-dimensional items, such that the overall cost is minimized. Thus, both DSP and DST are NP-hard, already for a single good, and hard to approximate for arbitrary number of goods.In this paper we focus on finding efficient approximations for DSP and DST instances where the number of goods is some fixed constant. In particular, we develop polynomial time approximation schemes (PTAS) for several subclasses of instances of practical interest. Our results include a PTAS for CIP in fixed dimension, and a more efficient (combinatorial) scheme for CIP_∞, where the multiplicity constraints are omitted. Our approximation scheme for CIP_∞ is based on a non-trivial application of the fast scheme for the fractional covering problem, proposed by Fleischer [L. Fleischer, A fast approximation scheme for fractional covering problems with variable upper bounds, in: Proc. of the 15th ACM-SIAM Symposium on Discrete Algorithm, 2004, pp. 994-1003].     

Accuracy evaluation of unsteady CFD numerical schemes by vortex preservation

Numerical uncertainty is an important but sensitive subject in computational fluid dynamics and there is a need for improved methods to quantify calculation accuracy. A known analytical solution, a Lamb-type vortex unsteady movement in a free stream, is compared to the numerical solutions obtained from different numerical schemes to assess their temporal accuracies. Solving the Navier-Stokes equations and using the standard Linearized Block Implicit ADI scheme, with first order accuracy in time second order in space, a vortex is convected and results show the rapid diffusion of the vortex. These calculations were repeated with the iterative implicit ADI scheme which has second-order time accuracy. A considerable improvement was noticed. The results of a similar calculation using an iterative fifth-order spatial upwind-biased scheme is also considered. The findings of the present paper demonstrate the needs and provide a means for quantification of both distribution and absolute values of numerical error.     

Uniform and optimal schemes for stiff initial-value problems

We formulate a class of difference schemes for stiff initial-value problems, with a small parameter ε multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ε, that is the solution of the difference scheme u_i^h satisfies |u_i^h−u(x_i)| Ch, where C is independent of h and ε. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |u_i^h−u(x_i)| C min (h, ε). Finally, we show that this class contains higher-order schemes.     

A relaxed block-triangular splitting preconditioner for generalized saddle-point problems

For the generalized saddle-point problems, based on a new block-triangular splitting of the saddle-point matrix, we introduce a relaxed block-triangular splitting preconditioner to accelerate the convergence rate of the Krylov subspace methods. This new preconditioner is easily implemented since it has simple block structure. The spectral property of the preconditioned matrix is analysed. Moreover, the degree of the minimal polynomial of the preconditioned matrix is also discussed. Numerical experiments are reported to show the preconditioning effect of the new preconditioner.     

Approximate response sensitivities for nonlinear problems in explicit dynamic formulation

Explicit and implicit time integration schemes are discussed in the context of sensitivity analysis of dynamic problems. The application of the fully explicit central difference method (CDM) proves to be efficient for many nonlinear problems. In the case of the corresponding dynamic sensitivity problem the CDM is less advantageous both from efficiency and accuracy points of view. Approximate sensitivity expressions are derived in the paper for nonlinear path-dependent problems allowing the application of an unconditionally stable implicit time integration scheme with the time step much larger than the time step of the explicit CDM scheme of the direct problem. The method seems to be particularly suitable for problems of quasi-static nature in which the dynamic terms are artificially introduced to allow explicit CDM solution of highly nonlinear equations. Received January 21, 2000     

An explicit scheme for the solution of the filtration problems

Different models of compressible fluid filtration are considered. Unlike the classical system of equations, the continuity equation is modified with allowance for the minimum scale of space averaging and for the internal relaxation time of the system. Three-level explicit finite difference schemes are proposed that are convenient for high-performance parallel implementation. The transition from the parabolic to the hyperbolic system of equations makes the stability requirements for them less stringent than for the two-level schemes.     

Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.