Well-posedness and Stability of Exact Non-reflecting Boundary Conditions

Eriksson, Sofia; Nordström, Jan

doi:10.2514/6.2013-2960

Cited by 1 publication

(2 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fully discrete SBP-SAT approximation of (4.1) can now be written as 10) where the initial and boundary conditions are imposed by…”

Section: First Derivative Approximationsmentioning

confidence: 99%

“…Proposition 4.1 opens up the possibility for using more general nonreflecting boundary conditions (which typically contain time derivatives) and prove stability by using the energy method. Normally one has to use normal mode analysis, which is significantly more complicated, and essentially limited to one dimensional problems, see [9,10].…”

Section: First Derivative Approximationsmentioning

confidence: 99%

See 1 more Smart Citation

Summation-By-Parts in Time: The Second Derivative

Nordström¹,

Lundquist²

2016

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates, are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type including time and space derivatives.We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including non-reflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil.Key words. time integration, second derivative approximation, initial value problem, high order accuracy, wave equation, second order form, initial boundary value problems, boundary conditions, stability, convergence, finite difference, summation-by-parts operators, weak initial conditions AMS subject classifications. 65L20,65M061. Introduction. Hyperbolic partial differential equations on second order form appear in many fields of applications including electromagnetics, acoustics and general relativity, see for example [1,17] and references therein. The most common way of solving these equations has traditionally been to rewrite them on first order form and apply well-established methods for solving first order hyperbolic problems. For the time integration part, a popular choice due to its time reversibility is the leapfrog method, especially for long time simulations of wave propagation problems. This traditional approach does however have some disadvantages. Most notably it increases the number of unknowns and requires a higher resolution in both time and space [16]. Many attempts have been made to instead discretize the second order derivatives directly with finite difference approximations. Various types of compact difference schemes have been proposed, e.g. including the classical methods of Störmer and Numerov [11,21,13], as well as one-step methods of Runge-Kutta type [11].Of particular interest to us is the development of schemes based on high order accurate operators in space obeying a summation-by-parts (SBP) rule together with the weak simultaneous-approximation-term (SAT) penalty technique for boundary conditions, [7,15]. The SBP-SAT technique in space in combination with well posed boundary conditions leads to energy stable semi-discrete schemes. By augmenting the SBP-SAT technique in space with SBP-SAT in time, we arrive at a procedure that leads in an almost automatic way to fully discrete schemes with uncon...

show abstract

“…The fully discrete SBP-SAT approximation of (4.1) can now be written as 10) where the initial and boundary conditions are imposed by…”

Section: First Derivative Approximationsmentioning

confidence: 99%

Section: First Derivative Approximationsmentioning

confidence: 99%