On the order of accuracy for difference approximations of initial-boundary value problems

Svärd, Magnus; Nordström, Jan

doi:10.1016/j.jcp.2006.02.014

Cited by 159 publications

(202 citation statements)

References 14 publications

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“…As time-integrator, the classical 4th-order Runge-Kutta method with 5000 grid points was used. The results shown in Table 1 confirm that the scheme is accurate for the 2nd-, 3rd-, 4th-and 5th-order SBP-SAT schemes [23].…”

Section: Rate Of Convergence For the Deterministic Casesupporting

confidence: 63%

“…We discretize using high-order finite difference methods on summation-by-parts form with weakly imposed boundary conditions, and prove strong stability [23,24]. The statistics of the solution such as the mean, variance and confidence intervals are computed non-intrusively using quadrature rules for the given stochastic distributions [10,12].…”

Section: Introductionmentioning

confidence: 99%

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The effect of uncertain geometries on advection–diffusion of scalar quantities

Wahlsten

Nordström

2017

Bit Numer Math

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The two dimensional advection-diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.

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Section: Rate Of Convergence For the Deterministic Casesupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

The effect of uncertain geometries on advection–diffusion of scalar quantities

Wahlsten

Nordström

2017

Bit Numer Math

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“…We will solve (13) using a semi-discrete finite di↵erence formulation based on the SBP-SAT technique [10,11,12,13]. The reader is referred to [2,3] for complete technical details.…”

Section: The Semi-discrete Formulationmentioning

confidence: 99%

Stochastic Galerkin Projection and Numerical Integration for Stochastic Systems of Equations

Wahlsten¹,

Nordström²

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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An incompletely parabolic system in three space dimensions with stochastic boundary and initial data is studied. The intrusive approach where we combine polynomial chaos with stochastic Galerkin projection is compared to the non-intrusive approach, where quadrature rules in combination with probability density functions of the prescribed uncertainties are used. The two methods are compared when calculating statistics for the compressible Navier-Stokes equations. As a measure of comparison, variance size, computational e ciency and accuracy are used.

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“…More details on this productive and well tested technique is given below. For a read-up, see [3], [11], [13], [14], [20], [19], [21], [15], [2], [5]. A recipe for constructing a stable and convergent scheme when using the SBP-SAT technique is to choose the so called penalty parameters such that an energy-estimate is obtained.…”

Section: Recipe For Constructing a Schemementioning

confidence: 99%

“…More details on the weak imposition of boundary and interface conditions using the SAT technique will be given below. For a read-up on this technique see [3], [11], [13], [14], [20], [19], [21], [15]. By multiplying (17) from the left with U T (P ⊗ I) we obtain…”

Section: Sbp Operators and Weak Non Characteristic Boundary Conditionsmentioning

confidence: 99%

Linear and Nonlinear Boundary Conditions for Wave Propagation Problems

Nordström

2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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We discuss linear and nonlinear boundary conditions for wave propagation problems. The concepts of well-posedness and stability are discussed by considering a specific example of a boundary condition occurring in the modeling of earthquakes. That boundary condition can be formulated in a linear and nonlinear way and implemented in a characteristic and non-characteristic way. These differences are discussed and the implications and difficulties are pointed out. Numerical simulations that illustrate the theoretical discussion are presented together with an application that show that the methodology can be used for practical problems.

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On the order of accuracy for difference approximations of initial-boundary value problems

Cited by 159 publications

References 14 publications

The effect of uncertain geometries on advection–diffusion of scalar quantities

The effect of uncertain geometries on advection–diffusion of scalar quantities

Stochastic Galerkin Projection and Numerical Integration for Stochastic Systems of Equations

Linear and Nonlinear Boundary Conditions for Wave Propagation Problems

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