Based　on　the　four-equation　model　of　the　elastic　fluid-conveying　straight　pipe,　the　axial　coupling　vibration　characteristics　with　water　hammer　effect　are　studied　in　this　paper.　Two　numerical　schemes　were　developed　by　combining　the　finite　integration　method　(FIM)　with　the　implicit　Euler　method　(IEM)　and　with　the　numerical　inversion　of　Laplace　transform　(NILT),　respectively,　to　efficiently　and　accurately　analyze　the　axial　coupling　vibration　of　the　pipe.　A　mixed　function,　consisting　of　the　radial　basis　function　(RBF)　and　the　polynomial　basis　function　(PBF),　is　adopted　to　construct　the　interpolation　function　of　the　FIM　at　each　node.　Then,　the　integral　value　of　a　function　can　be　discretized　into　linear　combinations　of　the　function　values　at　the　node.　This　feature　makes　the　FIM　easy　to　program,　straightforward　and　efficient　when　it　is　applied　to　a　complicated　computational　region.　To　evaluate　the　accuracy　and　robustness　of　the　proposed　scheme　for　dealing　with　the　discontinuous　wave　problems,　it　was　compared　with　other　numerical　methods　and　reasonable　agreements　were　achieved.　It　indicates　that　FIM　can　accurately　capture　the　nonlinear　process　with　water　hammer　effect.　Nevertheless,　on　processing　the　temporal　term,　the　numerical　oscillation　occurs　at　the　discontinuity　for　NILT　while　IEM　does　not　have.　Moreover,　when　considering　the　interaction　of　Poisson　and　connection　coupling,　the　pipe　pressure　wave　appears　to　have　some　delay　in　the　phase　compared　with　the　classic　water　hammer.　Due　to　the　superposition　of　the　pipe　vibration　and　the　forced　vibrations　produced　by　liquids,　the　pressure　fluctuation　increases　significantly　compared　with　the　situation　when　only　considering　Poisson　coupling.　©　2019,　Nanjing　Univ.　of　Aeronautics　an　Astronautics.　All　right　reserved.