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SUPERCONVERGENT FUNCTIONAL ESTIMATES FROM SUMMATIONBYPARTS Finitedifference Discretizations
, 2011
"... Diagonalnorm summationbyparts (SBP) operators can be used to construct timestable highorder accurate finitedifference schemes. However, to achieve both stability and accuracy, these operators must use sorder accurate boundary closures when the interior scheme is 2sorder accurate. The bounda ..."
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Diagonalnorm summationbyparts (SBP) operators can be used to construct timestable highorder accurate finitedifference schemes. However, to achieve both stability and accuracy, these operators must use sorder accurate boundary closures when the interior scheme is 2sorder accurate. The boundary closure limits the solution to (s + 1)order global accuracy. Despite this bound on solution accuracy, we show that functional estimates can be constructed that are 2sorder accurate. This superconvergence requires dualconsistency, which depends on the SBP operators, the boundary condition implementation, and the discretized functional. The theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension. In higher dimensions, we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces. The generality of the theoretical results is demonstrated using a twodimensional Poisson problem and a nonlinear hyperbolic system—the Euler equations of fluid mechanics.
Interface and Boundary Schemes for HighOrder Methods
, 2009
"... Highorder finitedifference methods show promise for delivering efficiency improvements in some applications of computational fluid dynamics. Their accuracy and efficiency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes offer several adv ..."
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Highorder finitedifference methods show promise for delivering efficiency improvements in some applications of computational fluid dynamics. Their accuracy and efficiency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes offer several advantages. In particular, they are an effective means of dealing with mesh nonsmoothness, which can arise from the geometry definition or mesh topology. In this paper, two such interface schemes are compared for a hyperbolic problem. Both schemes are stable and provide the required order of accuracy to preserve the desired global order. The first uses standard difference operators up to thirdorder global accuracy and special nearboundary operators to preserve stability for fifthorder global accuracy. The second scheme combines summationbyparts operators with simultaneous approximation terms at interfaces and boundaries. The results demonstrate the effectiveness of both approaches in achieving their prescribed orders of accuracy and quantify the error associated with the introduction of interfaces. Overall, these schemes offer several advantages, and the error introduced at mesh interfaces is small. Hence they provide a highly competitive option for dealing with mesh interfaces and boundary conditions in highorder multiblock solvers, with the summationbyparts approach with simultaneous approximation terms preferred for its more rigorous stability properties. I.
A parallel NewtonKrylov flow solver for the three dimensional Reynoldsaveraged NavierStokes equations
 In 20th Annual Conference CFD Society of
, 2012
"... This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are impl ..."
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Cited by 6 (3 self)
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This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are implemented using simultaneous approximation terms. Functionals, drag and lift, are calculated using Simpson’s rule. The solver is verified using the method of manufactured solutions and Ringleb flow and validated using the ONERA M6 wing. The results demonstrate that the combination of highorder finitedifference operators with a parallel NewtonKrylov solution technique is an excellent option for efficient computation of aerodynamic flows. I.
A parallel NewtonKrylovSchur flow solver for the NavierStokes equations using the SBPSAT approach
, 2010
"... This paper presents a threedimensional NewtonKrylov flow solver for the NavierStokes equations which uses summationbyparts (SBP) operators on multiblock structured grids. Simultaneous approximation terms (SAT’s) are used to enforce the boundary conditions and the coupling of block interfaces. ..."
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This paper presents a threedimensional NewtonKrylov flow solver for the NavierStokes equations which uses summationbyparts (SBP) operators on multiblock structured grids. Simultaneous approximation terms (SAT’s) are used to enforce the boundary conditions and the coupling of block interfaces. The discrete equations are solved iteratively with an inexact Newton method. The linear system of each Newton iteration is solved using a Krylov subspace iterative method with an approximateSchur parallel preconditioner. The algorithm is validated against an established twodimensional flow solver. Additionally, results are presented for laminar flow around the ONERA M6 wing, as well as low Reynolds number flow around a sphere. Using 384 processors, the solver is capable of obtaining the steadystate solution (reducing the flow residual by 12 orders of magnitude) on a 4.1 million node grid around the ONERA M6 wing in 4.2 minutes. Convergence to 3 significant figures in force coefficients is achieved in 83 seconds. Parallel scaling tests show that the algorithm scales well with the number of processors used. The results show that the SBPSAT discretization, solved with the parallel NewtonKrylovSchur algorithm, is an efficient option for threedimensional NavierStokes solutions, with the SAT’s providing several advantages in enforcing boundary conditions and block coupling. I.
Derivatives for TimeSpectral Computational Fluid Dynamics Using an Automatic Differentiation
 Adjoint,” AIAA Journal
, 2012
"... In computational fluid dynamics, for problems with periodic flow solutions, the computational cost of spectral methods is significantly lower than that of full, unsteady computations. As is the case for regular steadyflow problems, there are various interesting periodic problems, such as those inv ..."
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In computational fluid dynamics, for problems with periodic flow solutions, the computational cost of spectral methods is significantly lower than that of full, unsteady computations. As is the case for regular steadyflow problems, there are various interesting periodic problems, such as those involving helicopter rotor blades, wind turbines, or oscillating wings, that can be analyzed with spectral methods. When conducting gradientbased numerical optimization for these types of problems, efficient sensitivity analysis is essential. We develop an accurate and efficient sensitivity analysis for timespectral computational fluid dynamics. By combining the cost advantage of the spectralsolution methodology with an efficient gradient computation, we can significantly reduce the total cost of optimizing periodic unsteady problems. The efficient gradient computation takes the form of an automatic differentiation discrete adjoint method, which combines the efficiency of an adjoint method with the accuracy and rapid implementation of automatic differentiation. To demonstrate the method, we compute sensitivities for an oscillating ONERA M6 wing. The sensitivities are shown to be accurate to 8–12 digits, and the computational cost of the adjoint computations is shown to scale well up to problems of more than 41 million state variables. Nomenclature
Optimal diagonalnorm SBP operators
, 2013
"... Optimal boundary closures are derived for first derivative, finite difference operators of order 2, 4, 6 and 8. The closures are based on a diagonalnorm summationbyparts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multiblock grids and entropy stability for non ..."
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Optimal boundary closures are derived for first derivative, finite difference operators of order 2, 4, 6 and 8. The closures are based on a diagonalnorm summationbyparts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multiblock grids and entropy stability for nonlinear equations that support a convex extension. The new closures are developed by enriching conventional approaches with additional boundary closure stencils and nonequidistant grid distributions at the domain boundaries. Greatly improved accuracy is achieved near the boundaries, as compared with traditional diagonal norm operators of the same order. The superior accuracy of the new optimal diagonalnorm SBP operators is demonstrated for linear hyperbolic systems in one dimension and for the nonlinear compressible Euler equations in two dimensions. Key words: finite difference methods, multiblock grids, high order accuracy, stability, boundary treatment, Euler equations
Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models
"... This is a preprint of the following article, which is available from ..."
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This is a preprint of the following article, which is available from
Linköping University Electronic Press Report A Flexible Far Field Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain A Flexible Far Field Boundary Procedure for Hyper bolic Problems: Multiple Penalty Terms Applied in a
"... Abstract. A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summationbyparts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied at far field boundaries in an extended domai ..."
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Abstract. A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summationbyparts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied at far field boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct nonreflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.
Linköping University Post Print A stable and conservative method for locally adapting the design order of finite difference schemes A stable and conservative method for locally adapting the design order of finite difference schemes
, 2011
"... Abstract A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing SummationByParts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoret ..."
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Abstract A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing SummationByParts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable. Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.
SummationByParts Operators and HighOrder Quadrature ✩
"... Summationbyparts (SBP) operators are finitedifference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP d ..."
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Summationbyparts (SBP) operators are finitedifference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operatorusedforthequadrature. ThisquadraturehassignificantimplicationsforSBPbased discretizations; in particular, the diagonal norm accurately approximates the L2 norm for functions, and multidimensional SBP discretizations accurately approximate the divergence theorem.