| 1. |
- Jacobs, Byron A., et al.
(författare)
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On the order reduction of approximations of fractional derivatives: an explanation and a cure
- 2023
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Ingår i: BIT Numerical Mathematics. - : SPRINGER. - 0006-3835 .- 1572-9125. ; 63:1
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Tidskriftsartikel (refereegranskat)abstract
- Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss.
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| 2. |
- Laurén, Fredrik, 1990-, et al.
(författare)
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Energy stable wall modeling for the Navier-Stokes Equations
- 2021
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Close to solid walls, at high Reynolds numbers, fluids may develop steep gradients which requirea 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.
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| 3. |
- Laurén, Fredrik, 1990-, et al.
(författare)
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Energy stable wall modeling for the Navier-Stokes equations
- 2022
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 457
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Laurén, Fredrik, 1990-, et al.
(författare)
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Practical Inlet Boundary Conditions for Internal Flow Calculations
- 2018
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Ingår i: Computers & Fluids. - : Elsevier. - 0045-7930 .- 1879-0747. ; 175, s. 159-166
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Laurén, Fredrik, 1990-, et al.
(författare)
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Spectral properties of the incompressible Navier-Stokes equations
- 2021
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Ingår i: Journal of Computational Physics. - : Academic Press Inc.. - 0021-9991 .- 1090-2716. ; 429
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Tidskriftsartikel (refereegranskat)abstract
- The influence of different boundary conditions on the spectral properties of the incompressible Navier-Stokes equations is investigated. By using the Fourier-Laplace transform technique, we determine the spectra, extract the decay rate in time, and investigate the dispersion relation. In contrast to an infinite domain, where only diffusion affects the convergence, we show that also the propagation speed influence the rate of convergence to steady state for a bounded domain. Once the continuous equations are analyzed, we discretize using high-order finite-difference operators on summation-by-parts form and demonstrate that the continuous analysis carries over to the discrete setting. The theoretical results are verified by numerical experiments, where we highlight the necessity of high accuracy for a correct description of time-dependent phenomena.
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| 6. |
- Laurén, Fredrik, 1990-
(författare)
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Summation-by-parts formulations for flow problems
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Many problems in engineering and physics can be described by partial differential equations (PDEs). Augmented with proper initial and boundary conditions, the PDE forms an initial-boundary value problem (IBVP). An IBVP is said to be well-posed if a unique solution bounded by the given data exists. The behavior of the solution as well as the well-posedness of the IBVP are heavily influenced by the form and position of the boundary conditions. The solution, if it exists, to an IBVP cannot in general be obtained in closed form. Instead, one can compute an approximation using numerical methods. Summation-by-parts (SBP) operators together with the simultaneous approximation term (SAT) technique can be used to form stable numerical methods that generate accurate approximations. All discretizations in this thesis are based on the SBP-SAT framework.The first part of the thesis concentrates on IBVPs describing fluid motion. Different sets of boundary conditions are investigated in terms of energy-boundedness and spectral properties. Wall models are also studied and used to aid coarse-grid simulations in an energy stable manner.In the second part of the thesis, special discretization operators are derived. First, single-block SBP operators are combined with interpolation operators. The result is a single SBP operators on a multi-block grid that encapsulates both the metric terms and the interface treatments. Second, new SBP operators are developed for an application in high-energy physics. The new operators preserves the unit-trace, which is important since it enables a probability interpretation of the density matrix.
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| 7. |
- Lundquist, Tomas, 1986-, et al.
(författare)
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A multi-domain summation by-parts formulation for complex geometries
- 2021
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We combine existing discretization methods to obtain a simplified numerical formulation of partial derivative equations posed on multi-block/element domains. The interfaces (conforming or non-conforming) between blocks are treated with inner-product-preserving interpolation operators. The result is a single set of high-order multi-block summation-by-parts operators that encapsulates both the metric terms and the interface treatments. This enables for a clean description of the numerical scheme that mimics the essential features of its continuous counterpart. Furthermore, the stability analysis on a multi-block domain is simplified both for linear and nonlinear equations, since no problem-specific interface conditions must be derived and implemented. The theoretical developments are valid for all discretization methods that are included in the summation-by-parts framework. We exemplify the combined operator technique by considering a nonlinearly stable discrete formulation of the incompressible Navier-Stokes equations and perform calculations on an underlying multiblock domain.
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| 8. |
- Lundquist, Tomas, 1986-, et al.
(författare)
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A multi-domain summation-by-parts formulation for complex geometries
- 2022
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Ingår i: Journal of Computational Physics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9991 .- 1090-2716. ; 463
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Tidskriftsartikel (refereegranskat)abstract
- We combine existing summation-by-parts discretization methods to obtain a simplified numerical framework for partial differential equations posed on complex multi-block/element domains. The interfaces (conforming or non-conforming) between blocks are treated with inner-product-preserving interpolation operators, and the result is a high-order multi-block operator on summation-by-parts form that encapsulates both the metric terms as well as the interface treatments. This enables for a compact description of the numerical scheme that mimics the essential features of its continuous counterpart. Furthermore, the stability analysis on a multi-block domain is simplified for both for linear and nonlinear equations, since no problem-specific interface conditions need to be derived and implemented. We exemplify the combined operator technique by considering a nonlinearly stable discrete formulation of the incompressible Navier-Stokes equations and perform calculations on an underlying multi-block domain.
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| 9. |
- Nchupang, Mojalefa P., et al.
(författare)
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A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations
- 2023
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Ingår i: Computers & Fluids. - : PERGAMON-ELSEVIER SCIENCE LTD. - 0045-7930 .- 1879-0747. ; 267
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Tidskriftsartikel (refereegranskat)abstract
- In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.
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| 10. |
- Nordström, Jan, 1953-, et al.
(författare)
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A stable and conservative nonlinear interface coupling for the incompressible Euler equations
- 2022
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Ingår i: Applied Mathematics Letters. - : Elsevier. - 0893-9659 .- 1873-5452. ; 132
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Tidskriftsartikel (refereegranskat)abstract
- Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. By discretely mimicking the continuous analysis using summation-by-parts operators, we prove that the numerical scheme is stable and conservative. The theoretical findings are verified by numerical experiments.
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