| 1. |
- La Cognata, Cristina, 1985-
(författare)
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High order summation-by-parts based approximations for discontinuous and nonlinear problems
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained.The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials.In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions.The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied.
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| 2. |
- Erickson, Brittany A., et al.
(författare)
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Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems
- 2014
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 271, s. 328-338
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Tidskriftsartikel (refereegranskat)abstract
- Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal and spatial scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θθ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.
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| 3. |
- Eriksson, Sofia, et al.
(författare)
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Finite difference schemes with transferable interfaces for parabolic problems
- 2018
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Ingår i: Journal of Computational Physics. - Linköping : Elsevier. - 0021-9991 .- 1090-2716. ; 375, s. 935-949
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Tidskriftsartikel (refereegranskat)abstract
- We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent. (C) 2018 Elsevier Inc. All rights reserved.
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| 4. |
- Frenander, Hannes, et al.
(författare)
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Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative.
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A one dimensional steady-state advection-diffusion problem using summation-by-parts operators has been investigated. For approximating the second derivative, a wide stencil has been used, which has spurious, oscillating, modes for all mesh-sizes. We show that the size of the spurious modes are equal to the size of the truncation error for a stable approximation. The theoretical results are veried with numerical experiments.
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| 5. |
- Ghasemi Zinatabadi, Fatemeh, 1988-, et al.
(författare)
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An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations
- 2019
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Ingår i: Journal of Computational Physics. - Linköping : Linköping University Electronic Press. - 0021-9991 .- 1090-2716. ; 396, s. 280-302
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background ow with zero velocity normal to the interface.The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates.We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.
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| 6. |
- Ghasemi Zinatabadi, Fatemeh, 1988-, et al.
(författare)
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On conservation and dual consistency for summation-by-parts based approximations of parabolic problems
- 2019
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider the coupling of parabolic problems discretized using difference operators on summation-by-parts (SBP) form with interface conditions imposed weakly. In, it was shown that conservation and dual consistency are equivalent concepts for linear conservation laws. Here, we show that these concepts are equivalent also for symmetric or symmetrizable parabolic problems, exemplified by the heat equation. We rewrite the heat equation as first order system as in the local discontinuous Galerkin method and show the equivalence of dual consistency and conservation for both the first and second order forms.
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| 7. |
- Ghasemi Zinatabadi, Fatemeh, 1988-, et al.
(författare)
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The Relation Between Primal and Dual Boundary Conditions for Hyperbolic Systems of Equations
- 2019
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problems. In particular, we show that the primal and dual boundary conditions are related by a simple scaling relation. It is also shown that the weak dual problem can be derived directly from the weak primal problem.Based on the continuous analysis, we discretize and perform computations with a high-order finite difference scheme on summation-by-parts form with weak boundary conditions. It is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems. Numerical experiments corroborate the theoretical results.
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| 8. |
- Kopriva, David A., et al.
(författare)
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Error boundedness of discontinuous Galerkin spectral element approximations of hyperbolic problems
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.
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| 9. |
- Kreiss, Gunilla, et al.
(författare)
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Elimination of first order errors in shock calculations
- 2001
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM PUBLICATIONS. - 0036-1429 .- 1095-7170. ; 38:6, s. 1986-1998
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Linders, Viktor, et al.
(författare)
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Accurate solution-adaptive finite difference schemes for coarse and fine grids
- 2020
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 410
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Tidskriftsartikel (refereegranskat)abstract
- We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.
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