Showing papers on "Finite difference method published in 2002"
TL;DR: An efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three‐dimensional lattice and a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies.
Abstract: This article describes a number of algorithms that are designed to improve both the efficiency and accuracy of finite difference solutions to the Poisson-Boltzmann equation (the FDPB method) and to extend its range of application. The algorithms are incorporated in the DelPhi program. The first algorithm involves an efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three-dimensional lattice. This method constitutes a significant improvement over existing methods in terms of its combination of speed and accuracy. The DelPhi program has also been expanded to allow the definition of geometrical objects such as spheres, cylinders, cones, and parallelepipeds, which can be used to describe a system that may also include a standard atomic level depiction of molecules. Each object can have a different dielectric constant and a different surface or volume charge distribution. The improved definition of the surface leads to increased precision in the numerical solutions of the PB equation that are obtained. A further improvement in the precision of solvation energy calculations is obtained from a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies. The program allows for finite difference grids of large dimension; currently a maximum of 571 3 can be used on molecules containing several thousand atoms and charges. As described elsewhere, DelPhi can also treat mixed salt systems containing mono- and divalent ions and provide electrostatic free energies as defined by the nonlinear PB equation.
682 citations
585 citations
TL;DR: An efficient implementation of the finite difference Poisson–Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems.
Abstract: We report here an efficient implementation of the finite difference Poisson–Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems. This is achieved by implementing the algorithm with Eisenstat's two optimizations, utilizing the electrostatic update in simulations, and applying prudent approximations, including: relaxing the convergence criterion, not updating Poisson–Boltzmann-related forces every step, and using electrostatic focusing. It is also possible to markedly accelerate the supporting routines that are used to set up the calculations and to obtain energies and forces. The resulting finite difference Poisson–Boltzmann method delivers efficiency comparable to the distance-dependent dielectric model for a system tested, HIV Protease, making it a strong candidate for solution-phase molecular dynamics simulations. Further, the finite difference method includes all intrasolute electrostatic interactions, whereas the distance dependent dielectric calculations use a 15-A cutoff. The speed of our numerical finite difference method is comparable to that of the pair-wise Generalized Born approximation to the Poisson–Boltzmann method. © 2002 Wiley Periodicals, Inc. J Comput Chem 23: 1244–1253, 2002
429 citations
TL;DR: This work investigates the application of a high-order finite difference method for compressible large-eddy simulations on stretched, curvilinear and dynamic meshes and finds the compact/filtering approach to be superior to standard second and fourth-order centered, as well as third-order upwind-biased approximations.
Abstract: This work investigates the application of a high-order finite difference method for compressible large-eddy simulations on stretched, curvilinear and dynamic meshes. The solver utilizes 4th and 6th-order compact-differencing schemes for the spatial discretization, coupled with both explicit and implicit time-marching methods. Up to 10th order, Pade-type low-pass spatial filter operators are also incorporated to eliminate the spurious high-frequency modes which inevitably arise due to the lack of inherent dissipation in the spatial scheme. The solution procedure is evaluated for the case of decaying compressible isotropic turbulence and turbulent channel flow. The compact/filtering approach is found to be superior to standard second and fourth-order centered, as well as third-order upwind-biased approximations. For the case of isotropic turbulence, better results are obtained with the compact/filtering method (without an added subgrid-scale model) than with the constant-coefficient and dynamic Smagorinsky models. This is attributed to the fact that the SGS models, unlike the optimized low-pass filter, exert dissipation over a wide range of wave numbers including on some of the resolved scales
423 citations
TL;DR: In this paper, a stabilized finite element method is proposed to solve the transient Navier-Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales.
406 citations
TL;DR: In this paper, a moving boundary technique was developed to investigate wave runup and rundown with depth-integrated equations using a high-order finite difference scheme, which is used to solve highly nonlinear and weakly dispersive equations.
350 citations
TL;DR: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network by constraining the form of the velocity profile across the vessel radius.
Abstract: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network. By constraining the form of the velocity profile across the vessel radius, the three-dimensional Navier--Stokes equations are reduced to one-dimensional equations governing conservation of mass and momentum. These equations are coupled to a pressure-radius relationship characterizing the elasticity of the vessel wall to describe the transient blood flow through a vessel segment. The two step Lax--Wendroff finite difference method is used to numerically solve these equations. The flow through bifurcations, where three vessel segments join, is governed by the equations of conservation of mass and momentum. The solution to these simultaneous equations is calculated using the multidimensional Newton--Raphson method. Simulations of blood flow through a geometric model of the coronary network are presented demonstrating phy...
329 citations
TL;DR: It is concluded that holes in the skull can be treated reliably by means of the BEM and should be incorporated in forward and inverse modeling.
Abstract: Holes in the skull may have a large influence on the EEG and ERP. Inverse source modeling techniques such as dipole fitting require an accurate volume conductor model. This model should incorporate holes if present, especially when either a neuronal generator or the electrodes are close to the hole, e.g., in case of a trephine hole in the upper part of the skull. The boundary element method (BEM) is at present the preferred method for inverse computations using a realistic head model, because of its efficiency and availability. Using a simulation approach, we have studied the accuracy of the BEM by comparing it to the analytical solution for a volume conductor without a hole, and to the finite difference method (FDM) for one with a hole. Furthermore, we have evaluated the influence of holes on the results of forward and inverse computations using the BEM. Without a hole and compared to the analytical model, a three-sphere BEM model was accurate up to 5-10%, while the corresponding FDM model had an error <0.5%. In the presence of a hole, the difference between the BEM and the FDM was, on average, 4% (1.3-11.4%). The FDM turned out to be very accurate if no hole is present. We believe that the difference between the BEM and the FDM represents the inaccuracy of the BEM. This inaccuracy in the BEM is very small compared to the effect that holes can have on the scalp potential (up to 450%). In regard to the large influence of holes on forward and inverse computations, we conclude that holes in the skull can be treated reliably by means of the BEM and should be incorporated in forward and inverse modeling.
321 citations
TL;DR: In this paper, a method of thin wire representation for the FDTD method that is suitable for the three-dimensional surge simulation is presented, which is indispensable to simulate electromagnetic surges on wires or steel flames of which the radius is smaller than a discretized space step.
Abstract: Simulation of very fast surge phenomena in a three-dimensional structure requires a method based on Maxwell's equations such as the finite difference time domain (FDTD) method or the method of moments (MoM), because circuit-equation-based methods cannot handle the phenomena. This paper presents a method of thin wire representation for the FDTD method that is suitable for the three-dimensional surge simulation. The thin wire representation is indispensable to simulate electromagnetic surges on wires or steel flames of which the radius is smaller than a discretized space step used in the FDTD simulation. Comparisons between calculated and laboratory-test results are presented to show the accuracy of the proposed thin wire representation, and the development of a general surge analysis program based on the FDTD method is also described in the present paper.
298 citations
TL;DR: In this paper, large-eddy simulations of spatially developing planar turbulent jets are performed using a compact finite-difference scheme of sixth-order and an advective upstream splitting method-based method of second-order accuracy.
264 citations
TL;DR: It was possible to speed up solution of the bidomain equations by an order of magnitude with a slight decrease in accuracy, and direct methods were faster than iterative methods by at least 50% when a good estimate of the extracellular potential was required.
Abstract: The bidomain equations are the most complete description of cardiac electrical activity. Their numerical solution is, however, computationally demanding, especially in three dimensions, because of the fine temporal and spatial sampling required. This paper methodically examines computational performance when solving the bidomain equations. Several techniques to speed up this computation are examined in this paper. The first step was to recast the equations into a parabolic part and an elliptic part. The parabolic part was solved by either the finite-element method (FEM) or the interconnected cable model (ICCM). The elliptic equation was solved by FEM on a coarser grid than the parabolic problem and at a reduced frequency. The performance of iterative and direct linear equation system solvers was analyzed as well as the scalability and parallelizability of each method. Results indicate that the ICCM was twice as fast as the FEM for solving the parabolic problem, but when the total problem was considered, this resulted in only a 20% decrease in computation time. The elliptic problem could be solved on a coarser grid at one-quarter of the frequency at which the parabolic problem was solved and still maintain reasonable accuracy. Direct methods were faster than iterative methods by at least 50% when a good estimate of the extracellular potential was required. Parallelization over four processors was efficient only when the model comprised at least 500 000 nodes. Thus, it was possible to speed up solution of the bidomain equations by an order of magnitude with a slight decrease in accuracy.
TL;DR: A semi-implicit numerical model for the 3D Navier-Stokes equations on unstructured grids is derived and discussed in this article, where the governing differential equations are discretized by means of a finite difference-finite volume algorithm which is robust, very efficient, and applies to barotropic and baroclinic, hydrostatic and nonhydrostatic, and one-, two-, and three-dimensional flow problems.
TL;DR: In this paper, the authors present a rationale for the success of nonoscillatory finite volume difference schemes in modeling turbulent flows without need of subgrid scale models, and demonstrate that these truncation terms have physical justification, representing the modifications to the governing equations that arise when one considers the motion of finite volumes of fluid over finite intervals of time.
Abstract: We present a rationale for the success of nonoscillatory finite volume (NFV) difference schemes in modeling turbulent flows without need of subgrid scale models. Our exposition focuses on certain truncation terms that appear in the modified equation of one particular NFV scheme, MPDATA. We demonstrate that these truncation terms have physical justification, representing the modifications to the governing equations that arise when one considers the motion of finite volumes of fluid over finite intervals of time.
TL;DR: In this paper, the theoretical basis and the numerical implementation of free-vortex filament methods are reviewed for application to the prediction and analysis of helicopter rotor wakes, with a discussion of finite difference approximations to these equations and various numerical solution techniques.
Abstract: The theoretical basis and the numerical implementation of free-vortex filament methods are reviewed for application to the prediction and analysis of helicopter rotor wakes. The governing equations for the problem are described, with a discussion of finite difference approximations to these equations and various numerical solution techniques. Both relaxation and time-marching wake solution techniques are reviewed. It is emphasized how the careful consideration of stability and convergence (grid-independent behavior) are important to ensure a physically correct wake solution. The implementation of viscous diffusion and filament straining effects are also discussed. The need for boundary condition corrections to compensate for the inevitable wake truncation are described. Algorithms to accelerate the wake solution using velocity field interpolation are shown to reduce computational costs without a loss of accuracy. Several challenging examples of the application of free-vortex filament methods to helicopter rotor problems are shown, including multirotor configurations, flight near the ground, maneuvering flight conditions, and descending flight through the vortex ring state
TL;DR: In this paper, a numerical model based on the finite difference method is presented to predict tool and chip temperature fields in continuous machining and time varying milling processes, and the model is extended to milling where the cutting is interrupted and the chip thickness varies with time.
Abstract: In this paper, a numerical model based on the finite difference method is presented to predict tool and chip temperature fields in continuous machining and time varying milling processes. Continuous or steady state machining operations like orthogonal cutting are studied by modeling the heat transfer between the tool and chip at the tool—rake face contact zone. The shear energy created in the primary zone, the friction energy produced at the rake face—chip contact zone and the heat balance between the moving chip and stationary tool are considered. The temperature distribution is solved using the finite difference method. Later, the model is extended to milling where the cutting is interrupted and the chip thickness varies with time. The time varying chip is digitized into small elements with differential cutter rotation angles which are defined by the product of spindle speed and discrete time intervals. The temperature field in each differential element is modeled as a first-order dynamic system, whose time constant is identified based on the thermal properties of the tool and work material, and the initial temperature at the previous chip segment. The transient temperature variation is evaluated by recursively solving the first order heat transfer problem at successive chip elements. The proposed model combines the steady-state temperature prediction in continuous machining with transient temperature evaluation in interrupted cutting operations where the chip and the process change in a discontinuous manner. The mathematical models and simulation results are in satisfactory agreement with experimental temperature measurements reported in the literature.
TL;DR: General results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi- Bellman equations with variable coecients are obtained using systematically a tricky idea of N.V. Krylov.
Abstract: Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.
TL;DR: In this paper, a nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated, where the fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility.
TL;DR: Li and Yu as discussed by the authors developed a three-dimensional numerical model based on the full Navier-Stokes equations (NSE) in σ-coordinate to simulate two-dimensional solitary waves propagating in constant depth.
Abstract: A three-dimensional numerical model based on the full Navier–Stokes equations (NSE) in σ-coordinate is developed in this study. The σ-coordinate transformation is first introduced to map the irregular physical domain with the wavy free surface and uneven bottom to the regular computational domain with the shape of a rectangular prism. Using the chain rule of partial differentiation, a new set of governing equations is derived in the σ-coordinate from the original NSE defined in the Cartesian coordinate. The operator splitting method (Li and Yu, Int. J. Num. Meth. Fluids 1996; 23: 485–501), which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified NSE. The model is first tested for mass and energy conservation as well as mesh convergence by using an example of water sloshing in a confined tank. Excellent agreements between numerical results and analytical solutions are obtained. The model is then used to simulate two- and three-dimensional solitary waves propagating in constant depth. Very good agreements between numerical results and analytical solutions are obtained for both free surface displacements and velocities. Finally, a more realistic case of periodic wave train passing through a submerged breakwater is simulated. Comparisons between numerical results and experimental data are promising. The model is proven to be an accurate tool for consequent studies of wave-structure interaction. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: A general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes for weak solutions containing nonclassical regularization-sensitive shock waves.
Abstract: We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].
TL;DR: The aerodynamic generation of sound during phonation was studied using direct numerical simulations of the airflow and the sound field in a rigid pipe with a modulated orifice to find the dominant sound production mechanism was a dipole induced by the net force exerted by the surfaces of the glottis walls on the fluid along the direction of sound wave propagation.
Abstract: The aerodynamic generation of sound during phonation was studied using direct numerical simulations of the airflow and the sound field in a rigid pipe with a modulated orifice. Forced oscillations with an imposed wall motion were considered, neglecting fluid–structure interactions. The compressible, two-dimensional, axisymmetric form of the Navier–Stokes equations were numerically integrated using highly accurate finite difference methods. A moving grid was used to model the effects of the moving walls. The geometry and flow conditions were selected to approximate the flow within an idealized human glottis and vocal tract during phonation. Direct simulations of the flow and farfield sound were performed for several wall motion programs, and flow conditions. An acoustic analogy based on the Ffowcs Williams–Hawkings equation was then used to decompose the acoustic source into its monopole, dipole, and quadrupole contributions for analysis. The predictions of the farfield acoustic pressure using the acoustic...
TL;DR: In this article, the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials is reviewed and extended for nonorthogonal, nonsmooth, structured and unstructured computational grids.
Abstract: This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.
TL;DR: A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting and the effects of the noises on the accuracy of the approximations are illustrated.
Abstract: This paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises. A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting. Convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided.
TL;DR: In this paper, a 3D frequency-domain solution based on a volume integral equation approach has been implemented to simulate induction log responses arising from deviated boreholes intersecting horizontal bed boundaries.
Abstract: A 3‐D frequency‐domain solution based on a volume integral equation approach has been implemented to simulate induction log responses. In our treatment of the problem, we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy. The solution process uses a Krylov subspace iteration to solve the scattering equation, which is based on the modified iterative dissipative method. Internal consistency checks and comparisons with mode matching and finite‐difference solutions for vertical borehole models demonstrate the accuracy of the solution.There are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries. To simulate such responses requires the numerical solution of Maxwell's equations in three dimensions along with independent tests to validate the solution approach and its accuracy. In this paper, we compare two independent 3‐D frequency‐domain solutions for the ...
TL;DR: In this article, a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid was developed for the computation of the non-Newtonian extra-stress components on rigid boundaries.
Abstract: This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid. The basic equations governing the flow of an Oldroyd-B fluid are considered. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The resulting governing equations are solved by a finite difference method on a staggered grid, influenced by the ideas of the marker-and-cell (MAC) method. Numerical results demonstrating the capabilities of this new technique are presented for a number of problems involving unsteady free surface flows.
TL;DR: In this paper, an adaptive upwind finite-difference method based on high-order Weighted Essentially Non-Oscillatory (WENO) Runge-Kutta difference schemes for the paraxial eikonal equation is proposed.
Abstract: The point-source traveltime field has an upwind singularity at the source point. Consequently, all formally high-order, finite-difference eikonal solvers exhibit first-order convergence and relatively large errors. Adaptive upwind finite-difference methods based on high-order Weighted Essentially NonOscillatory (WENO) Runge-Kutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on a posteriori error estimation. It achieves prescribed accuracy at a far lower cost than does the fixed-grid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as take-off angles and geometric spreading factors.
TL;DR: In this paper, the second-order nonstandard finite differences (NSFDs) were used to improve the performance of the Yee algorithm for Mie scattering on a coarse grid.
Abstract: We previously described a high-accuracy version of the Yee algorithm that uses second-order nonstandard finite differences (NSFDs) and demonstrated its accuracy numerically. We now prove that at fixed frequency and grid spacing h, the leading error term is O(h/sup 6/) versus O(h/sup 2/) for the ordinary Yee algorithm with standard finite differences (SFDs). We numerically verify the superior accuracy of the NSFD algorithm by simulating near-field Mie scattering on a coarse grid and comparing with the SFD one and with analytical solutions. We present an updated stability analysis and show that the maximum time step for the NSFD algorithm is 20% longer than the SFD time step in two dimensions, and 36% longer in three dimensions. Finally, parameters that were previously given numerically are now analytically defined.
TL;DR: In this paper, a 3D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy.
Abstract: A 3-D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite-difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl-free and divergence-free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.
TL;DR: In this article, a singular mapping for nonlinear degenerate parabolic convection-diffusion equations is proposed, where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate.
Abstract: We analyze approximate solutions generated by an upwind difference scheme (of Engquist-Osher type) for nonlinear degenerate parabolic convection-diffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation u Δ , which is a manifestation of resonance. To circumvent this analytical prob- lem, we construct a singular mapping Ψ(γ,·) such that the total variation of the transformed variable z Δ =Ψ ( γ Δ ,u Δ ) can be bounded uniformly in Δ. This establishes strong L 1 com- pactness of z Δ and, since Ψ(γ,·) is invertible, also u Δ . Our singular mapping is novel in that it incorporates a contribution from the diffusion function A(u). We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kruzkov-type entropy inequality. We prove that the diffusion function A(u )i s Hcontin- uous, implying that the constructed weak solution u is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed.
TL;DR: The second order maximum principle preserving finite difference scheme for linear parabolic equations, using the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives is developed.
Abstract: New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.
TL;DR: A Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws and is more cost effective than the Runge--KuttaTime discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
Abstract: In this paper we develop a Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge--Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff time discretization procedure is more cost effective than the Runge--Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.