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Close to solid walls, at high Reynolds numbers, fluids may develop steep gradients which require a fine mesh for an accurate simulation of the turbulent boundary layer. An often used cure is to use a wall model instead of a fine mesh, with the drawback that modeling is introduced, leading to possibly unstable numerical schemes. In this paper, we leave the modeling aside, take it for granted, and propose a new set of provably energy stable boundary procedures for the incompressible Navier-Stokes equations. We show that these new boundary procedures lead to numerical results with high accuracy even for coarse meshes where data is partially obtained from a wall model. Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an energy bound but the linearised version does not (or vice versa). In this paper we attempt to resolve that contradiction and relate nonlinear energy conserving and bounded initial boundary value problems to their linearised versions and the related dual problems. We start by showing that a specific skew-symmetric form of the primal nonlinear problem leads to energy conservation and a bound. Next, we show that this specific form together with a non-standard linearisation procedure preserves these properties for the new slightly modified linearised problem. We proceed to show that the corresponding linear and nonlinear dual (or self-adjoint) problems also have bounds and conserve energy due to this specific formulation. Next, the implication of the new formulation on the choice of boundary conditions is discussed. A straightforward nonlinear and linear analysis may lead to a different number and type of boundary conditions required for an energy bound. We show that the new formulation sheds somelight on this contradiction. We conclude by illustrating that the new continuous formulation automatically leads toenergy stable and energy conserving numerical approximations for both linear and nonlinear primal and dual problems if the approximations are formulated on summation-by-parts form. Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier–Stokes equations. Funding agencies: European Commission [ACP3-GA-2013-605036]

To impose boundary conditions, data at the boundaries must be known, and consequently measurements of the imposed quantities must be available. In this paper, we consider the two most commonly used inflow boundary conditions with available data for internal flow calculations: the specification of the total temperature and total pressure. We use the energy method to prove that the specification of the total temperature and the total pressure together with the tangential velocity at an inflow boundary lead to well-posedness for the linearized compressible Euler equations. Next, these equations are discretized in space using high-order finite-difference operators on summation-by-parts form, and the boundary conditions are weakly imposed. The resulting numerical scheme is proven to be stable and the implementation of the corresponding nonlinear scheme is verified with the method of manufactured solutions. We also derive the spectrum for the continuous and discrete problems and show how to predict the convergence rate to steady state. Finally, nonlinear steady-state computations are performed, and they confirm the predicted convergence rates.

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability are analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to the coupling of fluid-acoustic models, multi-grid implementations, adaptive mesh refinements, multi-block formulations and numerical filtering. Funding agencies: Swedish Meteorological and Hydrological Institute (SMHI)

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved. Funding agencies: VINNOVA, the Swedish Governmental Agency for Innovation Systems [2013-01209]

Affordable, high order simulations of turbulent flows on unstructured grids for very high Reynolds number flows require wall models for efficiency. However, different wall models have different accuracy and stability properties. Here, we develop a kinetic energy stability estimate to investigate stability of wall model boundary conditions. Using this norm, two wall models are studied, a popular equilibrium stress wall model, which is found to be unstable and the dynamic slip wall model which is found to be stable. These results are extended to the discrete case using the Summation by parts (SBP) property of the discontinuous Galerkin method. Numerical tests show that while the equilibrium stress wall model is accurate but unstable, the dynamic slip wall model is inaccurate but stable., Accepted in Computers and Fluids

The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception. We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data. The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.

Unresolved gradients produce numerical oscillations and inaccurate results. The most straightforward solution to such a problem is to increase the resolution of the computational grid. However, this is often prohibitively expensive and may lead to ecessive execution times. By training a neural network to predict the shape of the solution, we show that it is possible to reduce numerical oscillations and increase both accuracy and efficiency. Data from the neural network prediction is imposed using multiple penalty terms inside the domain.

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.

n/a Classified in the journal as "Short note"

We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic approach that rests on the well-established theory of RK methods. The purpose of this paper is to provide a bottom-up overview of stability and convergence results for linear and non-linear problems that relate to general RK-SBP methods. To this end, we focus on the RK viewpoint, since this perspective so far is largely unexplored. This approach allows us to derive all results as simple consequences of the properties of SBP methods combined with well-known results from RK theory. In this way, new proofs of known results such as A-, L- and B-stability are given. Additionally, we establish previously unreported results such as strong S-stability, dissipative stability and stiff accuracy of certain RK-SBP methods. Further, it is shown that a subset of methods are B-convergent for strictly contractive non-linear problems and convergent for non-linear problems that are both contractive and dissipative.

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.

We construct accurate central difference stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh, and with an easily obtained bound on the dispersion error. This is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a Chebyshev set. In this new formulation, characterising and numerically obtaining optimised schemes can be done using established theory.

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

We consider a hyperbolic system with uncertainty in the boundary and initial data. Our aim is to show that different boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As applications, we study the effect of this technique on Maxwell's equations as well as on a subsonic outflow boundary for the Euler equations.

A detailed account of the stability and accuracy properties of the SBP-SAT technique for numerical time integration is presented. We show how the technique can be used to formulate both global and multi-stage methods with high order of accuracy for both stiff and non-stiff problems. Linear and non- linear stability results, including A-stability, L-stability and B-stability are proven using the energy method for general initial value problems. Numerical experiments corroborate the theoretical properties.

In this paper we derive new far-field boundary conditions for the time-dependent Navier-Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedness of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier-Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations. We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

The stochastic Galerkin and collocation methods are used to solve an advection–diffusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection–diffusion equation onto the stochastic basis functions. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steady-state.

In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier–Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters. The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to almost full slip. Original Publication:Jens Berg and Jan Nordström, Stable Robin solid wall boundary conditions for the Navier-Stokes equations, 2011, Journal of Computational Physics, (230), 19, 7519-7532.http://dx.doi.org/10.1016/j.jcp.2011.06.027Copyright: Elsevierhttp://www.elsevier.com/

Simulations of the interaction between a vortex and a NACA0012 airfoil are performed with a stable, high-order accurate (in space and time), multi-block finite difference solver for the compressible Navier–Stokes equations. We begin by computing a benchmark test case to validate the code. Next, the flow with steady inflow conditions are computed on several different grids. The resolution of the boundary layer as well as the amount of the artificial dissipation is studied to establish the necessary resolution requirements. We propose an accuracy test based on the weak imposition of the boundary conditions that does not require a grid refinement. Finally, we compute the vortex–airfoil interaction and calculate the lift and drag coefficients. It is shown that the viscous terms add the effect of detailed small scale structures to the lift and drag coefficients. Original Publication:Magnus Svärd, Johan Lundberg and Jan Nordström, A computational study of vortex-airfoil interaction using high-order finite difference methods, 2010, Computers & Fluids, (39), 1267-1274.http://dx.doi.org/10.1016/j.compfluid.2010.03.009Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/

This paper analyzes well-posedness and stability of a conjugate heat transfer problem in one space dimension. We study a model problem for heat transfer between a fluid and a solid. The energy method is used to derive boundary and interface conditions that make the continuous problem well-posed and the semi-discrete problem stable. The numerical scheme is implemented using 2nd-, 3rd- and 4th-order finite difference operators on Summation-By-Parts (SBP) form. The boundary and interface conditions are implemented weakly. We investigate the spectrum of the spatial discretization to determine which type of coupling that gives attractive convergence properties. The rate of convergence is verified using the method of manufactured solutions. Original Publication:Jens Lindström and Jan Nordström, A stable and high-order accurate conjugate heat transfer problem, 2010, Journal of Computational Physics, (229), 5440-5456.http://dx.doi.org/10.1016/j.jcp.2010.04.010Copyright: Elsevier Science B.V., Amsterdamhttp://www.elsevier.com/

The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to inuence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.

Investigation of acceleration effects on missile aerodynamics using computational fluid dynamics

Node-centred edge-based finite volume approximations are very common in computational fluid dynamics since they are assumed to run on structured, unstructured and even on mixed grids. We analyse the accuracy properties of both first and second derivative approximations and conclude that these schemes cannot be used on arbitrary grids as is often assumed. For the Euler equations first-order accuracy can be obtained if care is taken when constructing the grid. For the Navier-Stokes equations, the grid restrictions are so severe that these finite volume schemes have little advantage over structured finite difference schemes. Our theoretical results are verified through extensive computations.

A stable wall boundary procedure is derived for the discretized compressible Navier-Stokes equations. The procedure leads to an energy estimate for the linearized equations. We discretize the equations using high-order accurate finite difference summation-by-parts (SBP) operators. The boundary conditions are imposed weakly with penalty terms. We prove linear stability for the scheme including the wall boundary conditions. The penalty imposition of the boundary conditions is tested for the flow around a circular cylinder at Ma=0.1 and Re=100. We demonstrate the robustness of the SBP-SAT technique by imposing incompatible initial data and show the behavior of the boundary condition implementation. Using the errors at the wall we show that higher convergence rates are obtained for the high-order schemes. We compute the vortex shedding from a circular cylinder and obtain good agreement with previously published (computational and experimental) results for lift, drag and the Strouhal number. We use our results to compare the computational time for a given for a accuracy and show the superior efficiency of the 5th-order scheme.

In this paper, we present a stable hybrid scheme for viscous problems. The hybrid method combines the unstructured finite volume method with high-order finite difference methods on complex geometries. The coupling procedure between the two numerical methods is based on energy estimates and stable interface conditions are constructed. Numerical calculations show that the hybrid method is efficient and accurate.

We construct a stable high-order finite difference scheme for the compressible Navier-Stokes equations, that satisfy an energy estimate. The equations are discretized with high-order accurate finite difference methods that satisfy a Summation-By-Parts rule. The boundary conditions are imposed with penalty terms known as the Simultaneous Approximation Term technique. The main result is a stability proof for the full three-dimensional Navier-Stokes equations, including the boundary conditions. We show the theoretical third-, fourth-, and fifth-order convergence rate, for a viscous shock, where the analytic solution is known. We demonstrate the stability and discuss the non-reflecting properties of the outflow conditions for a vortex in free space. Furthermore, we compute the three-dimensional vortex shedding behind a circular cylinder in an oblique free stream for Mach number 0.5 and Reynolds number 500.

New sets of boundary conditions for the velocity-pressure formulation of the incompressible Navier-Stokes equations are derived. The boundary conditions have the same form on both inflow and outflow boundaries and lead to a divergence free solution. Moreover, the specific form of the boundary condition makes it possible derive a symmetric positive definite equation system for the internal pressure. Numerical experiments support the theoretical conclusions.

A high-order accurate finite difference scheme is used to perform numerical studies on the benefit of high-order methods. The main advantage of the present technique is the possibility to prove stability for the linearized Euler equations on a multi-block domain, including the boundary conditions. The result is a robust high-order scheme for realistic applications. Convergence studies are presented, verifying design order of accuracy and the superior efficiency of high-order methods for applications dominated by wave propagation. Furthermore, numerical computations of a more complex problem, a vortex-airfoil interaction, show that high-order methods are necessary to capture the significant flow features for transient problems and realistic grid resolutions. This methodology is easy to parallelize due to the multi-block capability. Indeed, we show that the speedup of our numerical method scales almost linearly with the number of processors.

High order finite difference approximations are derived for the second order wave equation with discontinuous coefficients, on rectangular geometries. The discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion. Each sub-domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method. This guarantees a conservative, strictly stable and high order accurate scheme. The analysis is verified by numerical simulations in one and two spatial dimensions.

Our objective is to derive stable first-, second- and fourth-order artificial dissipation operators for node based finite volume schemes. Of particular interest are general unstructured grids where the strength of the finite volume method is fully utilised.A commonly used finite volume approximation of the Laplacian will be the basis in the construction of the artificial dissipation. Both a homogeneous dissipation acting in all directions with equal strength and a modification that allows different amount of dissipation in different directions are derived. Stability and accuracy of the new operators are proved and the theoretical results are supported by numerical computations.

Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and 2nd-order hyperbolic, partial differential equations are considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundary without reducing the global order of accuracy.This result is generalised to initial-boundary value problems with an mth-order principal part. Then, the boundary accuracy can be lowered m orders.Further, it is shown that schemes using summation-by-parts operators that approximate second derivatives are pointwise bounded. Linear and nonlinear computations, including the two-dimensional Navier-Stokes equations, corroborate the theoretical results.

A stable hybrid method for hyperbolic problems that combines the unstructured finite volume method with high-order finite difference methods has been developed. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical calculations verify that the hybrid method is efficient and accurate.

We discuss how to combine the node based unstructured finite volume method widely used to handle complex geometries and nonlinear phenomena with very efficient high order finite difference methods suitable for wave propagation dominated problems. This fully coupled numerical procedure reflects the coupled character of the sound generation and propagation problem. The coupling procedure is based on energy estimates and stability can be guaranteed. Numerical experiments using finite difference methods that shed light on the theoretical results are performed. To cite this article: J. Nordstrom, J. Gong, C. R. Mecanique 333 (2005).

Finite difference operators approximating second derivatives and satisfying a summation by parts rule have been derived for the fourth, sixth and eighth order case by using the symbolic mathematics software Maple. The operators are based on the same norms as the corresponding approximations of the first derivative, which makes the construction of stable approximations to general parabolic problems straightforward. The error analysis shows that the second derivative approximation can be closed at the boundaries with an approximation two orders less accurate than the internal scheme, and still preserve the internal accuracy.

The unstructured node centered finite volume method is analyzed and it is shown that it can be interpreted in the framework of summation by parts operators. It is also shown that introducing boundary conditions weakly produces strictly stable formulations. Numerical experiments corroborate the analysis.

The often-used practice of extrapolating all variables at a subsonic, outflow boundary is investigated. For steady state calculations, we show that the L2 error in a subdomain of fixed size decreases with the distance to the far field boundary. Thus, error reduction can be obtained by expanding the size of the computational domain. Numerical experiments using the Euler equations corroborate the theoretical prediction.

Corrigendum to “A stable and conservative interface treatment of arbitrary spatial accuracy” [J.Comput.Phys.148(1999)341–365]

High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates

Strictly stable finite volume formulations for long time integration of hyperbolic problems are formulated by modifying conventional and widely used finite volume schemes close to the boundary. The modification leads to difference operators that satisfy a summation-by-parts rule and the boundary conditions are imposed by a penalty procedure. Both node centered and cell centered approximations are considered. Numerical studies corroborate the superior stability of the modified formulations for long time integrations.

One way of imposing boundary conditions for the Navier–Stokes equations at artificial supersonic outflow boundaries is to use flux-extrapolation. This procedure is analyzed and compared with variable-extrapolation both for the semi-discrete and continuous problem using the Laplace transform technique. It is shown that flux-extrapolation leads to a loss of accuracy and an extra time growth. The numerical experiments support the analytical results.

Stable and accurate interface conditions based on the SAT penalty method are derived for the linear advection?diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP) property. We prove that stability is a natural consequence of the SBP operators used in conjunction with the new, penalty type, boundary conditions. In addition, we show that the interface treatments are conservative. The issue of the order of accuracy of the interface boundary conditions is clarified. New finite-difference operators of spatial accuracy up to sixth order are constructed which satisfy the SBP property. These finite-difference operators are shown to admit design accuracy (pth-order global accuracy) when (p?1)th-order stencil closures are used near the boundaries, if the physical boundary conditions and interface conditions are implemented to at leastpth-order accuracy. Stability and accuracy are demonstrated on the nonlinear Burgers' equation for a 12-subdomain problem with randomly distributed interfaces.

Boundary and interface conditions for high-order finite difference methods applied to the constant coefficient Euler and Navier?Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.

A very common procedure when constructing boundary conditions for the time-dependent Navier-Stokes equations at artificial boundaries is to extrapolate the solution from grid points near the boundary to the boundary itself. For supersonic outflow, where all the characteristic variables leave the computational domain, this leads to accurate results. In the case of subsonic outflow, where one characteristic variable enters the computational domain, one cannot in general expect accurate solutions by this procedure. The problem with outflow boundary operators of extrapolation type at artificial boundaries with errors in the boundary data of order one will be investigated. Both the problem when the artificial outflow boundary is located in essentially uniform flow and the situation when the artificial outflow boundary is located in a flow field with large gradients are discussed. It will be shown, that in the special case when there are large gradients tangential to the boundary, extrapolation methods can be used even in the subsonic case.

The use of characteristic boundary conditions for the Navier-Stokes equations with constant coefficients matrices are analysed. New boundary conditions of characteristic type that yield fast convergence to steady state, a strongly well posed continuous problem and a strongly stable semi-discrete problem are derived. Numerical experiments that indicate the relevance of the analysis for the nonlinear problem are presented.