Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.     

Numerical investigation of fully developed channel flow using shock-capturing schemes

The ability to simulate wall-bounded channel flows with second- and third-order shock-capturing schemes is tested on both subsonic and supersonic flow regimes, respectively at Mach 0.5 and 1.5. Direct numerical simulations (DNSs) and large-eddy simulations (LESs) are performed at Reynolds number 3000.In both flow regimes, results are compared with well-documented DNS, LES or experimental data.At Ma₀=0.5, a simple second-order centred scheme provides results in excellent agreement with incompressible DNS databases, while the addition of artificial or subgrid-scale (SGS) dissipation decreases the resolution accuracy giving just satisfactory results. At Ma₀=1.5, the second-order space accuracy is just sufficient to well resolve small turbulence scales on the chosen grid: without any dissipation models, such accuracy provides results in good agreement with reference data, while the addition of dissipation models considerably reduces the turbulence level and the flow appears almost laminar. Moreover, the use of explicit dissipative SGS models reduces the results accuracy.In both flow regimes, the numerical dissipation due to the discretization of the convective terms is also interpreted in terms of SGS dissipation in an LES context, yielding a generalised dynamic coefficient, equivalent to the dynamic coefficient of the Germano et al. [Phys. Fluids A 3(7) (1991) 1760] SGS model. This new generalised coefficient is thus developed to compare the order of magnitude of the intrinsic numerical dissipation of a shock-capturing scheme with respect to the SGS dissipation.     

A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains

A finite volume scheme which is based on fourth order accurate central differences in the spatial directions and on a hybrid explicit/semi-implicit time stepping scheme was developed to solve the incompressible Navier-Stokes equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity at the singularity of the cylindrical coordinate system and a new stability condition. The new method was applied in the direct numerical simulations (DNS) of the fully developed non-swirling turbulent flow through straight pipes with circular cross-section for the Reynolds number Re_τ = 360 based on the friction velocity u_τ and the pipe diameter. The obtained results are expressed in terms of statistical moments of the velocity components and are presented in comparison with those obtained with a second order accurate scheme and by measurements. It is shown that the fourth order spatial discretization leads to improved higher order statistical moments, while the first and the second order moments are more or less insensitive to the spatial discretization order.     

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge–Kutta methods for nonlinear problems and for linear problems as well as implicit Runge–Kutta methods and multi step methods will be collected     

High Order Strong Stability Preserving Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties—in any norm, seminorm or convex functional—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge–Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods. The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255. The work of D.I. Ketcheson was supported by a US Dept. of Energy Computational Science Graduate Fellowship under grant DE-FG02-97ER25308. The research of C.-W. Shu is supported in part by NSF grants DMS-0510345 and DMS-0809086.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank-Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.     

A Spectral Method for the Time Evolution in Parabolic Problems

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods     

A low-dissipation DG method for the under-resolved simulation of low Mach number turbulent flows

In recent years the use of high-order Discontinuous Galerkin (DG) methods for the under-resolved direct numerical simulations (uDNS) of turbulent flows has received special attention. The suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme: while the dispersion errors are small over a broad range of frequencies, a relevant dissipation error mainly acts at the smallest under-resolved scales, resembling a high frequency filter. Nevertheless, it was recognized (Flad and Gassner, 2017; Mengaldo et al., 2018) that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the uDNS approach. In this regard, the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided, in particular when dealing with low-speed flows, since this behaviour is exacerbated approaching the incompressibility limit. Fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods, see for example Weiss and Smith (1995). In this work a simple modification of the dissipation term of the low Mach preconditioned Roe scheme proposed by Weiss and Smith is considered. The aim is to reduce further the amount of numerical dissipation with the intent of improving the results of uDNS. A spatial DG discretization coupled with a linearly-implicit Rosenbrock-type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions. Results on canonical turbulent flow problems as the Taylor–Green vortex and the flow in a straight sided channel are presented. The improved accuracy of the proposed flux function is demonstrated. The new low-dissipation flux can be useful also in the context of standard, lower order, finite volume methods.     

Explicit Hybrid Time Domain Solver for the Maxwell Equations in 3D

We present an accurate and efficient explicit hybrid solver for Maxwell's equations in time domain. The hybrid solver combines FD-TD with an unstructured finite volume solver. The finite volume solver is a generalization of FD-TD to unstructured grids and it uses a third-order staggered Adams–Bashforth scheme for time discretization. A spatial filter of Laplace type is used by the finite volume solver to enable long simulations without suffering from late time instability problems. The numerical examples demonstrate that the hybrid solver is superior to stand-alone FD-TD in terms of accuracy and efficiency.     

High order parameter-robust numerical method for time dependent singularly perturbed reaction–diffusion problems

We introduce a high order parameter-robust numerical method to solve a Dirichlet problem for one-dimensional time dependent singularly perturbed reaction-diffusion equation. A small parameter ε is multiplied with the second order spatial derivative in the equation. The parabolic boundary layers appear in the solution of the problem as the perturbation parameter ε tends to zero. To obtain the approximate solution of the problem we construct a numerical method by combining the Crank–Nicolson method on an uniform mesh in time direction, together with a hybrid scheme which is a suitable combination of a fourth order compact difference scheme and the standard central difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or ε-uniform in the sense that its numerical solution converges to the exact solution uniformly well with respect to the singular perturbation parameter ε. More specifically, we prove that the numerical method is uniformly convergent of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.     

A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system

In this paper, a new implicit compact difference scheme is constructed for the fourth-order fractional diffusion-wave system by the method of order reduction. The temporal Caputo fractional derivative is discretized by an L₁ scheme. The spatial derivative of order 4 is reduced to one of order 2 by order reduction. Then, the reduced derivative of order 2 is discretized by a difference formula of order 4. Using order reduction, two simple and accurate formulae of discretization for the derivative boundary conditions are obtained. And a new way of proving the stability and convergence of the scheme is presented in this paper. Some numerical results demonstrate the accuracy and efficiency of our new scheme.     

An asymptotic estimate of the error of eigenvalues in the method of finite elements for the sturm-liouville problem

A scheme of the second order of accuracy for the Sturm—Liouville problem is constructed by the method of finite elements with the use of a special basic system of compact functions. The convergence of the method of finite elements is proved. An exact formula for estimating errors of eigenvalues is obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 28–36, March–Apni, 2000.     

High-resolution finite difference schemes using curvilinear coordinate grids for DNS of compressible turbulent flow over wavy walls

Direct numerical simulations of compressible turbulent flow over wavy wall geometries have been carried out by solving N–S equations on general curvilinear coordinates. A 6th order WENO scheme with minimized dispersion and controllable dissipation is employed to compute the inviscid fluxes, a 4th order central difference scheme is applied to compute the viscous fluxes, and a 6th order conservative compact scheme is used for computing the geometrical metrics. An implicit LU-SGS method is used for time integration to improve computational efficiency over the explicit schemes such as the Runge–Kutta approach. The validity and applicability of the present algorithm is confirmed by comparing our results with laboratory experimental measurements and DNS results in the literature.     

A numerical study of the convergence properties of ENO schemes

We report numerical results obtained with finite difference ENO schemes for the model problem of the linear convection equation with periodic boundary conditions. For the test function sin(x), the spatial and temporal errors decrease at the rate expected from the order of local truncation errors as the discretization is refined. If we take sin⁴(x) as our test function, however, we find that the numerical solution does not converge uniformly and that an improved discretization can result in larger errors. This difficulty is traced back to the linear stability characteristics of the individual stencils employed by the ENO algorithm. If we modify the algorithm to prevent the use of linearly unstable stencils, the proper rate of convergence is reestablished. The way toward recovering the correct order of accuracy of ENO schemes appears to involve a combination of fixed stencils in smooth regions and ENO stencils in regions of strong gradients —a concept that is developed in detail in a companion paper by Shu (this issue, 1990).     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.     

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.     

LES validation of turbulent rotating buoyancy- and surface tension-driven flow against DNS

The paper is concerned with the validation and error analysis of predictions for the flow and heat transfer in a silicon melt (Pr=0.013) found in a Czochralski (Cz) apparatus for crystal growth. This system resembles turbulent Rayleigh-Bénard-Marangoni convection. Since for practical applications predictions based on direct numerical simulations (DNS) require too many resources to conduct parametric studies or optimizations, nowadays in practice the method of choice is the large-eddy simulation (LES). The case considered consists of an idealized cylindrical crucible of 170 mm radius with a rotating crystal of 50 mm radius. Boundary conditions from experimental data were applied, which lead to the dimensionless numbers of , and Ra=2.8×10⁷. The filtered Navier-Stokes equations were solved based on a finite-volume scheme for curvilinear block-structured grids and an explicit time discretization. For a comprehensive error analysis, different grid sizes, subgrid-scale models, and discretization schemes were employed. The results were compared to reference DNS data of the same case recently generated by the authors (Int J Heat Mass Transfer, 51 (2008) 6219-6234) for validation. For the finest LES grid (10⁶ control volumes) using a standard Smagorinsky model with van Driest damping or a dynamic model, both with central discretization, the results agree well with the DNS reference while the computational effort could be reduced by a factor of 20. When using an upwind scheme even of formally second-order accuracy, significant deviations occur. Further stepwise reductions of the grid size decrease the CPU time drastically, but also lead to larger aberrations. When the grid is coarsened by a factor of 32 (resulting in ca. 130,000 CVs), even qualitative differences between the LES and the DNS solution appear.It could be shown in the present work that the LES method is an efficient tool to model the turbulent flow and heat transfer in Rayleigh-Bénard-Marangoni configurations. However, care should be taken in the choice of the grid resolution and discretization scheme for the nonlinear convective terms, as too coarse meshes in combination with upwind schemes lead to significant numerical errors. Finally, a quantified relation between the achievable accuracy and the necessary computational effort is presented.     

Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes

We study 2nd-, 4th-, 6th- and 8th-order accurate finite difference schemes approximating systems of conservation laws. Our goal is to utilize the high order of accuracy of the schemes for approximating complicated flow structures and add suitable diffusion operators to capture shocks. We choose appropriate viscosity terms and prove non-linear entropy stability. In the scalar case, entropy stability enables us to prove convergence to the unique entropy solution. Moreover, a limiter function that localizes the effect of the dissipation around discontinuities is derived. The resulting scheme is entropy stable for systems, and also converges to the entropy solution in the scalar case. We present a number of numerical experiments in order to demonstrate the robustness and accuracy of our scheme. The set of examples consists of a moving shock solution to the Burgers’ equation, a solution to the Euler equations that consists of a rarefaction and two contact discontinuities and a shock/entropy wave solution to the Euler equations (Shu’s test problem). Furthermore, we use the limited scheme to compute the solution to the linear advection equation and demonstrate that the limiter quickly vanishes for smooth flows and design/high-order of accuracy is retained. The numerical results in all experiments were very good. We observe a remarkable gain in accuracy when the order of the scheme is increased.     

Simulation of the Paraxial Laser Propagation Coupled with Hydrodynamics in 3D Geometry

We address here numerical simulation problems for modeling some phenomena arising in plasmas produced in experimental devices for Inertial Confinement Fusion. The model consists of a compressible fluid dynamics system coupled with a paraxial equation for modeling the laser propagation. For the fluid dynamics system, a numerical method of Lagrange–Euler type is used. For the paraxial equation, a time implicit discretization is settled which preserves the laser energy balance; the method is based on a splitting of the propagation term and the diffraction terms according to the propagation spatial variable. We give some features on the 3D implementation of the method in the parallel platform HERA. Results showing the accuracy of the numerical scheme are presented and we give also numerical results related to cases corresponding to realistic simulations, with a mesh containing up to 500 millions of cells.