Summation-by-parts in Time: the Second Derivative

• A new technique for time integration of initial value problems involving second derivatives is presented. The technique is based on summation-by-parts operators and weak initial conditions and lead to optimally sharp energy estimates. The schemes obtained in this way use wide operators, are unconditionally stable and high order accurate. The additional complications when using compact operators in time are discussed in detail and it is concluded that the existing compact formulations designed for space approximations are not appropriate. As an application we focus on the wave equation and derive optimal fully discrete energy estimates which lead to unconditional stability. 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 new methodology.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Time integration

second derivative approximations

initial value problems

high order accuracy

initial value boundary problems

boundary conditions

stability

convergence

summation-by-parts operators

Publikations- och innehållstyp

vet (ämneskategori)

rap (ämneskategori)