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A general interface procedure is presented for multi-domain collocation methods satisfying the summation-by-parts (SBP) spatial discretization convention. Unlike more traditional operators (e.g. FEM) applied to the advection-diffusion equation, the new procedure penalizes the solution and the first p derivatives across the interface. The combined interior/interface operators are proven to be pointwise stable, and conservative, although accuracy deteriorates for p≥2. Penalties between two different sets of variables are compared (motivated by FEM primal and flux formulations), and are shown to be equivalent for certain choices of penalty parameters. Extensive validation studies are presented using two classes of high-order SBP operators: (1) central finite difference, and (2) Legendre spectral collocation. The original publication is available at www.springerlink.com:Mark H. Carpenter, Jan Nordström and David Gottlieb, Revisiting and Extending Interface Penalties for Multidomain Summation-by-Parts Operators, 2010, Journal of Scientific Computing, (45), 118-150.http://dx.doi.org/10.1007/s10915-009-9301-5Copyright: Springer Science Business Mediahttp://www.springerlink.com/

In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems. The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case. The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach. Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks. Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability. Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

The stochastic 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 which gives a qualitative description of the development of the system over time for dierent initial and boundary conditions is presented. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Since the system is quite new, boundary conditions and related data are not known but has a weak connection to the original Burgers’ equation. Time-dependent boundary conditions require knowledge about the time development of high order moments of the boundary values. We will discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions.

The influence of weak and strong solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations

Analysis of mesh and boundary effects on the accuracy of node-centered finite volume schemes

One technique to measure mass flow in a microchannel is to measure the pressure drop over a well controlled orifice. Because of the small dimensions, the measurements are difficult to make precise. Manufacturing of MEMS components are expensive and time consumptive. It is therefore desirable with a computational model which can provide approximative but yet qualitative input on how to design the components. In this paper we address the problem of making a computational model of an orifice microchannel, perform computations using a high-order Navier-Stokes solver and evaluate the results using experimental data.

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately. Original Publication:Sofia Eriksson and Jan Nordström, Analysis of the order of accuracy for node-centered finite volume schemes, 2009, Applied Numerical Mathematics, (59), , 2659-2676.http://dx.doi.org/10.1016/j.apnum.2009.06.001Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/

We show how a stable and accurate hybrid procedure for fluid flow can be constructed. Two separate solvers, one using high order finite difference methods and another using the node-centered unstructured finite volume method are coupled in a truly stable way. The two flow solvers run independently and receive and send information from each other by using a third coupling code. Exact solutions to the Euler equations are used to verify the accuracy and stability of the new computational procedure. We also demonstrate the capability of the new procedure in a calculation of the flow in and around a model of a coral.

A stable and conservative high order multi-block method for the time-dependent compressible Navier-Stokes equations has been developed. Stability and conservation are proved using summation-by-parts operators, weak interface conditions and the energy method. This development makes it possible to exploit the efficiency of the high order finite difference method for non-trivial geometries. The computational results corroborate the theoretical analysis.

The Burgers' equation with uncertain initial and boundary conditions is investigated using a polynomial chaos (PC) expansion approach where the solution is represented as a truncated series of stochastic, orthogonal polynomials. The analysis of well-posedness for the system resulting after Galerkin projection is presented and follows the pattern of the corresponding deterministic Burgers equation. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Similarly to the deterministic case, the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix. The time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients. An estimate of the eigenvalues is provided. A characteristic analysis of the truncated PC system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions. It is shown that a precise statistical characterization of the input uncertainty is required and partial information, e.g. the expected values and the variance, are not sufficient to obtain a solution. An analytical solution is derived and the coefficients of the infinite PC expansion are shown to be smooth, while the corresponding coefficients of the truncated expansion are discontinuous.

In this paper well-posed boundary conditions for the Navier-Stokes equation are derived using the energy method. A new set of boundary conditions are compared with well-posed boundary conditions for the Euler equations when used to compute the subsonic flow over a flat plate. It is shown that the new set of boundary conditions are superior with regard to rate of convergence except for the highest Reynolds number tested. By combining the new boundary conditions with the Euler conditions the best overall performance was achieved.

Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.

In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

A mathematical tool box for the analytic treatment of non-stationary problems with relatively moving frames, including an extended vector analysis in n-dimensions, has been formulated by Lofgren’ , and forms the foundation for the finite volume formulation by Forsberg’. Consider a fixed, inertial reference frame X, referred to as the absolute frame (Fig. 1). Consider also a moving reference frame Z’, which may be accelerating in an arbitrary fashion, and which is referred to as the relative frame.

First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. Based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy.

To eliminate the problem with artificial boundary conditions and facilitate the use of Fourier methods, the fringe region (or filter, damping layer, absorbing layer, sponge layer) technique has been used in direct simulations of transitional and turbulent boundary layers. Despite the fact that good computational results have been obtained with this technique, it is not fully understood. The analysis in this paper indicates that the primary importance of the fringe region technique is to damp out the deviation associated with large scales in the direction normal to the wall. The lack of boundary conditions is compensated by the knowledge of an exact solution in the fringe region of the computational domain. The upstream influence from the fringe region is small. Numerical experiments verifying the theoretical predictions are presented.

The question of open boundary conditions of inflow and outflow type for the Navier-Stokes equations and its influence on the convergence to steady state is addressed. Both the continuous and semi-discrete problem are analysed using the energy method and the Laplace transform technique. The energy method is used to derive well-posed boundary conditions for the continuous problem. For the semi-discrete problem we use the energy method to prove that by using the well-posed boundary conditions for the continuous problem and adding a suitable numerical boundary condition well-posedness is preserved. By employing the Laplace transform technique the spectra for different types of boundary conditions are obtained. The spectra are analysed and it is shown how the choice of boundary conditions strongly affects the convergence to steady state. One-dimensional Navier-Stokes calculations are performed and the resulting convergence rates agree well with the theoretical analysis. Finally, the spectra obtained using inflow and outflow types of boundary conditions are compared with spectra obtained using periodic boundary conditions and the choice of a time-integration method for the Navier-Stokes equations is discussed.

We develop a general interface procedure to couple both structured and unstructured parts of a hybrid mesh in a non-collocated, multi-block fashion. The target is to gain optimal computational efficiency in fluid dynamics simulations involving complex geometries. While guaranteeing stability, the proposed procedure is optimized for accuracy and requires minimal algorithmic modifications to already existing schemes. Initial numerical investigations confirm considerable efficiency gains compared to non-hybrid calculations of up to an order of magnitude.

Two different edge-based spatial discretization schemes are investigated. In the first approach the discretization is applied to the edge control surface formed by the sum of the element sub-surfaces. In the second approach, the discretization is carried out on each sub- surface. It is shown theoretically that the discretization approach on the sub-surfaces may lead to enhanced accuracy at the expense of higher computational cost. The two approaches are applied in two dimensions over an airfoil and to the DLR F6 configuration. The accuracy is evaluated by conducting grid convergence studies. The discretization scheme on the sub- surfaces gives approximately first order accuracy in two dimensions, the other scheme is even less accurate. For the DLR F6 configuration in three dimensions, however, both approaches exhibit similar accuracy close to the expected second spatial order. 1 . Polar calculations and grid convergence studies were undertaken for the DLR-F6 wing-body (WB) and the WB configuration with a fairing (WB-FX2B) added to the junction between the wing trailing edge and fuselage. A summary of the results from the DPW3 contributions was given by Vassberg et al. 2,3 with a statistical analysis presented by Morrison and Hemsch 4 . The conclusion of that investigation was that, after removing outliers, both results with flow solvers using structured and unstructured grids give satisfactory grid convergence and hence accuracy. The most accurate results were obtained with finite volume solvers that rely on structured grids. The computed results on unstructured grids were slightly less accurate and, in particular, more sensitive to the selected family of grids. In this paper we investigate the grid convergence and solution accuracy of two different edge-based spatial discretization techniques in an unstructured, node-centered finite volume solver. The two approaches are compared primarily in terms of their grid convergence properties and hence solution accuracy but also steady state convergence and computational costs are discussed. They are applied to the WB configuration on three different groups of successively refined grids with different characteristics. The WB configuration was chosen since it has an area of flow separation at the wing-body trailing edge junction and the variation in the computed results of different participants was larger compared to the WB-FX2B where the flow is attached. In addition to the DLR F6 WB configuration, a similar grid convergence investigation is carried out over a two-dimensional airfoil. In the following section the flow solver with its solution algorithms is described together with the two discretization algorithms. The numerical accuracy of a convection and diffusion operator is investigated for a simple model grid in two dimensions. This is followed by the numerical results for the two test cases with concluding remarks at the end. II. Solution Algorithm The CFD solver employed in the calculations is the Edge code (http://www.edge.foi.se/), which is an edge- and node-based Navier-Stokes flow solver applicable for both structured and unstructured grids 5-7 . Edge is based on a finite-volume formulation where a median dual grid forms the control volumes with the unknowns allocated in the dual grid centres. The governing equations are integrated explicitly with a multi-stage Runge-Kutta scheme to

A line-implicit Runge-Kutta time stepping scheme is derived, implemented and applied. It is applied to fluid flow problems governed by the Navier-Stokes equations on stretched unstructured grids. The flow equations are integrated implicitly in time along structured lines in regions where the grid is stretched, typically in the boundary layer, and explicitly elsewhere. The integration technique is introduced for steady state problems with the intention to speed up the rate of convergence. It is extended to unsteady problems by a dual time stepping approach. The paper focuses on the implementation of the line-implicit scheme starting from an explicit multigrid flow solver and on the application of it. Numerical results are presented for test cases in two and three dimensions for inviscid and viscous flow problems. The line-implicit time integration convergence rates are compared to pure explicit convergence rates and the gain is quantified in terms of reduction of iterations and CPU time. All presented test cases show improved convergence rates. The gain is highest for the three dimensional test cases for which reductions of up to 75% of the computing time is obtained.

The problem with extrapolation boundary conditions at artificial outflow boundaries is investigated. A linear semi-discrete model problem describing the error introduced by the artificial boundary is analyzed. The effect of large gradients in the basic flow field is included in the model problem and appropriate error bounds are derived.

We show how the SBP−SAT technique can be used to derive a new family of methods for the time integration of initial value problems producing optimally sharp energy estimates. Some stability properties relevant for these methods are studied in detail, and accuracy results for both non-stiff and stiff problems are presented. We show that the technique is particularly suitable for the time integration of energy stable semi-discrete problems.

In this work, propagation of acoustic waves in the air intake of an turbo-fan engine are simulated using a commercial Navier-Stokes solver. Three different acoustic modes are studied, the first and ...