On Long Time Error Bounds for the Wave Equation on Second Order Form

Nordström, Jan; Frenander, Hannes

doi:10.1007/s10915-018-0667-0

Cited by 9 publications

(3 citation statements)

References 9 publications

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“…The boundedness of the norm of the error in the solutions U, V for the full system are shown in Figure 4. One can clearly see that the error is uniformly bounded, see also [56,57,58,59] for similar results.…”

Section: Numerical Resultssupporting

confidence: 60%

An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations

Ghasemi

Nordström

2019

Journal of Computational Physics

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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 flow 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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Section: Numerical Resultssupporting

confidence: 60%

An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations

Ghasemi

Nordström

2019

Journal of Computational Physics

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“…Energy estimates are useful for determining both location and number of boundary conditions needed to bound the solution, as well as provide a means for proving uniqueness of solutions to linear problems and insights into reasons for errorgrowth and error-boundedness, see [17,9,18]. In the section that follows we show existence (by construction) of a continuous solution to (1), and uniqueness and stability to perturbations in data follow from (5) or (7).…”

Section: The Scalar Equationmentioning

confidence: 94%

Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds

2019

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We derive analytic solutions to the scalar and vector advection equation with variable coe cients in one spatial dimension using Laplace transform methods. These solutions are used to investigate how accuracy and stability are influenced by the presence of discontinuous wave speeds when applying high-orderaccurate, skew-symmetric finite di↵erence methods designed for smooth wave speeds. The methods satisfy a summation-by-parts rule with weak enforcement of boundary conditions and formal order of accuracy equal to 2, 3, 4 and 5. We study accuracy, stability and convergence rates for linear wave speeds that are (a) constant, (b) non-constant but smooth, (c) continuous with a discontinuous derivative, and (d) constant with a jump discontinuity. Cases (a) and (b) correspond to smooth wave speeds and yield stable schemes and theoretical convergence rates. Nonsmooth wave speeds (cases (c) and (d)), however, reveal reductions in theoretical convergence rates and in the latter case, the presence of an instability.

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“…This procedure was developed in [31], where it was used to increase the convergence rate to steady state. It has also been used on linear problems to control error growth [34] and to aid coarsegrid computations [32]. In Article IV, we use the MPT technique together with data from the wall model to improve the coarse-grid results for the nonlinear INS equations.…”

Section: Article IVmentioning

confidence: 99%

Summation-by-parts formulations for flow problems

Laurén

2022

Linköping Studies in Science and Technology. Dissertations

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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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On Long Time Error Bounds for the Wave Equation on Second Order Form

Cited by 9 publications

References 9 publications

An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations

An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations

Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds

Summation-by-parts formulations for flow problems

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