A　long-time　stable　high-order　absorbing　boundary　condition　(ABC)　is　developed　for　the　finite　element　simulation　of　time-dependent　scalar　wave　propagation　in　unbounded　multilayer　media　that　is　a　multilayer　waveguide.　The　ABC　is　obtained　based　on　a　new　continued　fraction　(CF)　expansion　of　the　frequency-domain　dynamic　stiffness　matrix　on　the　artificial　boundary　of　a　truncated　infinite　domain.　The　dynamic　stiffness　in　scalar　CF　form　is　first　proposed　in　the　modal　space　for　a　single-layered　medium　and　is　subsequently　extended　to　a　matrix　CF　form　for　multilayer　media.　The　new　CF　converges　to　dynamic　stiffness　over　the　whole　frequency　range　as　its　order　increases　for　the　single-layered　medium,　and　has　high　accuracy　for　the　multilayer　media.　The　CF-based　high-order　ABC　is　therefore　accurate　and　stable　in　the　time　domain.　After　being　coupled　seamlessly　with　finite　element　method　(FEM),　instability　phenomena　are　observed　during　long-time　computation.　In　order　to　eliminate　the　instability　of　the　coupled　ABC　and　FEM　system,　a　straightforward　and　effective　method　is　proposed　by　introducing　damping　proportional　to　the　stiffness　matrix　of　spatially　discretised　finite　domain.　Numerical　examples　demonstrate　the　superior　properties　of　the　proposed　ABC　with　high　accuracy　and　long-time　stability　as　well　as　remarkable　coupling　with　FEM.　(C)　2018　Elsevier　B.V.　All　rights　reserved.