V N U Jo u rn al of Science, E arth Sciences 24 (2008) 10-15 Finite volume method for long wave runup: 1D model Phung Dang Hieu* Centerỷor Marine and Ocean-Atmosphere ỉnteraction Research Vietnam Institute of Meteorologỵ, Hydrology and Environment R cceived 20 D ecem ber 2007; receivcd in reviscd form 15 F eb ru ary 2008 A b stract A n u m erical m odcl using the 1D shallovv vvater e q u a tio n s vvas d evolopcd for the sim u la tio n o f long vvave p ro p ag atio n and ru n u p The d ev c lo p e d m odel is b a sc d on thc Finite V olum c M cth o d (FVM ) vvith an application oí G o d unov - ty p e sch e m e of socond o rd e r of accuracy T he m odel u s e s th e HLL ap p ro x im a te R iem ann solver for th e d e te rm in a tio n of n u m erical íluxcs at cell in teríaccs T h e m odel w as ap p lied to th c sim ulation of long w a v c p ro p a g atio n a n d ru n u p on a p lan e beach a n d sim u la tc d results w ere com pared w ith the p u b lish c d cx p erim en tal data The co m p ariso n sh o w s th a t th e p re sen t m odel has a povver of sim u la tio n o f long vvave p ro p a g atio n a n d ru n u p o n b eachcs Keỵivords: P inite V oỉum e M ethod; Shallovv W ater M odel; YVavc R unup In tro d u ctio n analytical resuits include tho one-dim ensional solution of C arrier and G reenspan (1958) for periodic vvave reílection from a plane beach [1] and the asym m etric solution by Thacker (1981) [6] for vvave resonance in a circular parabolic basin Synolakis (1987) [5] provided valuable experim ental data of long w ave ru n u p on a plane beach, w hich then vvere well k now n am ong Coastal engineering com m unity, w ho the job related with num erical m odeling oí Coastal hydrodynam ic processes A nalytical approach provides exact solution for idealized situation of geom etry and offers insights into the physical processes N um erical m odels provide approxim ate solutions in m ore general settings suitable for practical applications [9] Hovvever, the m ain challenge lies in the treatm ent of the m oving vvaterline and flow Long w ave ru n u p on beaches is one of tho hot challenging topics recently, for the ocean and Coastal engineering researchers Frequently, engineers face to problem s related to the sim u latio n or determ ination of w ave ru n u p in gcneral, and long vvave runup in particular for practical purposes, such as design of sea w all/ Coastal structures, etc Thereíore, d ev elo p m en t of a good m odel capable of sim u latio n of w ave ru n u p is vvorth for practical u sag e as vvell as for indoor researches Researchers have developed various analytical and num erical m odels based on the depth integrated shallovv vvater equations to explain the physical processes N otable discontinuity vvhon the w ater climbs u p and dovvn on beaches * Tel.: 84-4-7733090 E-mail: phungdanghieu@ \'kttv.edu.vn 10 Phung Dang tìieu / VN U Ịounuìl of Science, Earth Sciences 24 (2008) 10-15 So far, m any researchers have dcveloped m odels for the sim ulation of w ave runup Shuto and Goto (1978) [4] used íinite diííerence m ethod w ith a staggered schem e and a Lagrangian description of the m oving shoreline; Liu et al (1995) m odelcd ru n u p through ílooding and drying of the cells in response to adịacent w ater level changes [3] Titov and Synolakis (1995, 1998) [7, 8] proposed VTCS-2 m odel using the splitting technique and characteristic line m ethod Hu et al (2000) [2] developed an 1D m odel using FVM vvith a G odunov-type u p w in d scheme to sim ulate the w ave overtopping o í seawall VVei el al (2006) presented a m odcl for long w ave runup using exact Ricm ann solver [9] In this study, a num erical m odel is devclopcd using FVM and the robust approxim ate Riem ann solver HLL (H arten, Lax and van Leer) for the sim ulation of long wave ru n u p on a beach The m odel is veriíied for the case of experim ent p roposed by Synolakis (1987) C om parisons are carried out between sim ulated results and experim ental data (Synolakis, 1987) [5] The details of this study are given belovv N um erical m odel 2.1 Governing equation The present stu d y considers Onedim ensional (1D) depth-integrated Shallovv w ater equations in the C artesian coordinate system ( x ,t) The conservation form of the 1D non-linear shallovv w ater equations is vvritten as a u ỔF e — +— =s (1) õt õx where u is the vector of conserved variables; F is the ílux vectors; and s is the source term The explicit íorm of these vectors is explained 11 as follows: Hu (2 ) g H -A & p w here g : gravitational acceleration; p : w ater H iỉ+ ịg rí1 ,s = density; h : still vvater depth; H : total w ater depth, H = /i + ^ in vvhich Tj(x,t) is the displacem ent of w ater suríace ÍTom the still vvater level; Tx : bottom shear stress given by: Tx = p C ,u\u\,C f = g” (3) w here n: M anning coefficient for the bed roughness 2.2 Numerical scheme The íinite volum e íorm ulation im poses conservation laws in a control volume Integration of Eq (1) over a cell vvith the applieation of the G reen's theorem , gives: díì+ J.F-ndr= (4) w here í ì : cell dom ain; r : b o u n d ary of Q ; n : norm al outw ard vector of the boundary Taking the tim e integration of Eq (4) over duration Aí from to t2, w e have: r f r +jdf Ị F-nár =r |áíJ^SáQ (5) 'í í C onsidering the case of one-dim ensional m odel w ith cell size of A x , from Eq (5) we can deduce: — í ( x , / 2> — — AxAt i x