Results 1  10
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14
Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Differential Equations
, 1998
"... Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that ..."
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Cited by 55 (9 self)
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Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasionedimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multidimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations. Key words. PDEs, adjoint equations, error analysis, superconvergence AMS subject classifications. 65G99, 76N15 PII. S0036144598349423
A posteriori error estimates for higher order godunov finite volume methods on unstructured meshes
 Complex Applications III, R. Herbin and D. Kroner (Eds), HERMES Science Publishing Ltd
, 2002
"... A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise pth order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact ..."
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Cited by 19 (1 self)
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A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise pth order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact error representation formulas for derived functionals that are tailored to the class of high order Godunov finite volume methods with data reconstruction, R 0 pu0. We then devise computable error estimates that exploit the structure of Godunov finite volume methods. The present theory applies directly to a wide range of finite volume methods in current use including MUSCL, TVD, UNO, and ENO methods. Issues such as the treatment of nonlinearity and postprocessing of dual (adjoint) problem data are discussed. Numerical results for linear advection and nonlinear scalar conservation laws at steadystate are presented to validate the analysis.
Adaptive Lagrange–Galerkin methods for unsteady convectiondiffusion problems
, 2000
"... In this paper we derive an a posteriori error bound for the Lagrange–Galerkin discretisation of an unsteady (linear) convectiondiffusion problem, assuming only that the underlying spacetime mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated ..."
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Cited by 15 (7 self)
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In this paper we derive an a posteriori error bound for the Lagrange–Galerkin discretisation of an unsteady (linear) convectiondiffusion problem, assuming only that the underlying spacetime mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a userdefined tolerance.
Analytic adjoint solutions for the quasionedimensional Euler equations
 J. Fluid Mechanics
, 2001
"... The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is requir ..."
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Cited by 15 (6 self)
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The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock. 1.
Progress in adjoint error correction for integral functionals
 COMPUTING AND VISUALIZATION IN SCIENCE
, 2004
"... When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. an ..."
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Cited by 6 (0 self)
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When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. and the solution of an appropriately defined adjoint p.d.e. A number of applications are presented and the challenges of smooth reconstruction on unstructured grids and error correction for shocks are discussed.
On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control
 Comm. Numer. Meth. Eng
, 2008
"... Abstract. We study a recent timestep adaptation technique for hyperbolic conservation laws. The key tool is a spacetime splitting of adjoint error representations for target functionals due to Süli[19] and Hartmann[13]. It provides an efficient choice of timesteps for implicit computations of weakl ..."
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Cited by 4 (2 self)
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Abstract. We study a recent timestep adaptation technique for hyperbolic conservation laws. The key tool is a spacetime splitting of adjoint error representations for target functionals due to Süli[19] and Hartmann[13]. It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and become small when a perturbation enters the flow field. Besides using adjoint techniques which are already wellestablished, we also add a new ingredient which simplifies the computation of the dual problem. Due to Galerkin orthogonality, the dual solution ϕ does not enter the error representation as such. Instead, the relevant term is the difference of the dual solution and its projection to the finite element space, ϕ −ϕh. We can show that it is therefore sufficient to compute the spatial gradient of the dual solution, w = ∇ϕ. This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem, and in the same finite element space. We demonstrate the capabilities of the approach for a weakly instationary test problem for scalar conservation laws. Contents
A Posteriori Error Indicators for Hyperbolic Problems
, 1997
"... this paper we consider the design and implementation of an adaptive mesh refinement algorithm for the numerical solution of hyperbolic problems. In particular, emphasis will be given to the design of a local error indicator to identify regions in the computational domain where the error is locally l ..."
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Cited by 3 (0 self)
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this paper we consider the design and implementation of an adaptive mesh refinement algorithm for the numerical solution of hyperbolic problems. In particular, emphasis will be given to the design of a local error indicator to identify regions in the computational domain where the error is locally large. In practice, one of the most common approaches is to design the mesh according to the size of the local residual of the underlying partial differential operator. In this paper we show that the local residual on an element only controls a portion of the local error on , referred to as the
Adjoint Correction and Bounding of Error Using Lagrange Form of Truncation Term
 Computers & Mathematics with Applications
, 2005
"... An International Journal computers & ..."
Two a Posteriori Error Estimates for OneDimensional Scalar Conservation Laws
"... In this paper, we propose aposteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws. The main tool to derive them is a synthetic version of Kruzkov's estimates recently introduced by Bouchut and Perthame, further developed by Katsoulakis, Kos ..."
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Cited by 2 (0 self)
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In this paper, we propose aposteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws. The main tool to derive them is a synthetic version of Kruzkov's estimates recently introduced by Bouchut and Perthame, further developed by Katsoulakis, Kossioris and Makridakis. We first consider the schemes for which a consistent incell entropy inequality can be derived. Then, we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities. As an illustration, we show several numerical tests on the classical Burgers equation and we propose an adaptive algorithm for the selection of the mesh. Key words: conservation law, entropy dissipation, a posteriori error estimates, adaptive finite difference method. AMS subject classification: 65M15, 65M50. 1 Work partially supported by TMR project HCL N o . ERBFMRXCT960033 2 Foundation for Research and TechnologyHellas, Institute of Applied and Computat...
ADAPTIVE TIMESTEP CONTROL FOR INSTATIONARY SOLUTIONS OF THE EULER EQUATIONS
"... In this paper we continue our work on adaptive timestep control for weakly instationary problems [29]. The core of the method is a spacetime splitting of adjoint error representations for target functionals due to Süli [31] and Hartmann [18]. The main new ingredients are (i) the extension from sca ..."
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Cited by 1 (1 self)
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In this paper we continue our work on adaptive timestep control for weakly instationary problems [29]. The core of the method is a spacetime splitting of adjoint error representations for target functionals due to Süli [31] and Hartmann [18]. The main new ingredients are (i) the extension from scalar, 1D, conservation laws to the 2D Euler equations of gas dynamics, (ii) the derivation of boundary conditions for a new formulation of the adjoint problem and (iii) the coupling of the adaptive timestepping with spatial adaptation. For the spatial adaptation, we use a multiscalebased strategy developed by Müller [24], and we combine this with an implicit time discretization. The combined spacetime adaptive method provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and becomes small when a perturbation enters the flow field. The efficiency of the solver is investigated by means of an unsteady inviscid 2D flow over a bump.