Summation-By-Parts in Time: The Second Derivative
2016 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 3, A1561-A1586 p.Article in journal (Refereed) Published
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 that 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 nonreflecting 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.
Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2016. Vol. 38, no 3, A1561-A1586 p.
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
IdentifiersURN: urn:nbn:se:liu:diva-128697DOI: 10.1137/15M103861XOAI: oai:DiVA.org:liu-128697DiVA: diva2:931601