Suyeon Shin* and Woonjae Hwang**

Transcription

1 JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volme 5, No. 3, Agst THE NUMERICAL SOLUTION OF SHALLOW WATER EQUATION BY MOVING MESH METHODS Syeon Sin* and Woonjae Hwang** Abstract. Tis paper presents a moving mes metod for solving te yperbolic conservation laws. Moving mes metod consists of two independent parts: PDE evoltion and mes- redistribtion. We compte nmerical soltion of sallow water eqation by sing moving mes metods. In comparison wit comptations on a fied grid, te moving mes metod appears more accrate resoltion of discontinities.. Introdction Te moving mes metods ave been sed for solving differential eqations tat involve singlar soltion. It gives an improvement in accracy by moving mes points tat tey are concentrate in regions of larger soltion variations. Becase of tis advantage, many sefl scemes ave been applied on moving grids in recent years. Several moving mes tecniqes ave been introdced, in wic one of te most advocated metods is te one based on solving elliptic PDEs first proposed by Winslow [5]. Harten and Hyman [6] began te earliest stdy in tis direction, by moving te grid along te caracteristic direction to increase te accracy of soltion. After teir work, many oter moving mes metods ave been proposed. Salari and Steinberg [3] developed a FCT metod on a moving grid based on adaptive grid generation algoritms. Te moving mes metod based on armonic Received Jne 3, ; Accepted Jne 8,. Matematics Sbject Classification: Primary 65M5, 76M; Secondary 35L65. Key words and prases: moving mes, sallow water eqation, finite volme metods, conservation laws, Correspondence sold be addressed to Woonjae Hwang, woonjae@korea.ac.kr. **Spported by Basic Science Researc Program trog te National Researc Fondation of Korea(NRF) fnded by te Ministry of Edcation, Science and Tecnology(Grant No. -73).

2 564 Syeon Sin and Woonjae Hwang mapping was sggested by Dvinsky [4]. His metod can be viewed as a generalization and etension of Winslow s metod. Motivated by te work of Dvinsky, a moving mes finite element strategy based on armonic mapping was proposed and stdied by te Li et al. [9, ]. Te moving mes PDEs were stdied by Rssell et al. [7,, 6], and Li and Petzold []. In recent years, tere are many works abot moving mes metods [,, 3, 4, 5, 9,,,, 3, 4]. Adaptive mes metod consists of two independent parts [9, 9]: a mes- redistribtion algoritm and a soltion algoritm. Te first part is an iteration procedre. Meses are redistribted by an eqidistribtion principle. For te reslting elliptic, adaptive mes PDEs, a Gass-Seidel type iteration metod is sed. Te nderlying nmerical soltion on te new grids are pdated by a conservative-interpolation formla. Te second part will be independent of te first one, and it can be any of te standard codes for solving te given PDEs. Te organization of tis paper is as follows. In Section, te moving mes metod is briefly described. Nmerical eperiments of te onedimensional yperbolic conservation laws are provided in Section 3.. Moving mes metod In Tang and Tang [9, 4], te basic idea of te moving mes metod can be smmarized by two independent parts: mes-redistribtion and PDE evoltion... Mes-redistribtion based on Gass-Jacobi iteration Meses are redistribted by an eqidistribtion principle. Te mesredistribtion eqation is (.) (w ξ ) ξ =, < ξ < were te fnction w is called monitor fnction [6, 7], wic designed specifically to resolve discontinos soltions. To solve te mes redistribtion eqation (.), te following Gass-Jacobi type iteration can be sed: (.) w( j+ )( j+ j ) w( j )( j j ) =, and ten te nderlying nmerical soltion on te new grids { j+ } are pdated by a conservative-interpolation formla. Te conservativeinterpolation formla is following: (.3) j+ ũ j+ = j+ j+ ( j+ j ),

3 Te nmerical soltion by moving mes metods 565 were j+ = j+ j. Te linear fl c in (.3) is approimated by second-order nmerical fl. Te second-order nmerical fl is defined by (.4) j = c j (+ j + j ) c j (+ j j ). Te wave speed c j above is defined by c j = j j. In (.4), n,± j defined by are (.5) n,± j = n + j± ( j j± ) S j±, were S j+ by (.6) Sj+ wit is an approimation of te slope at j+. Sj+ = (sign( S + ) + sign( S )) j+ j+ S + S j+ j+ S + + S, j+ j+ is defined (.7) S+ j+ = n j+ 3 n j+ j+ 3 j+, S j+ = n n j+ j. j+ j Te eqation (.5) (.7) are te MUSCL(monotone pstream-centered sceme for conservation laws)-typed finite volme metod. Te new grid procedre by te Gass-Jacobi type iteration and te pdated soltion procedre sing te conservative-interpolation are repeated for a fied nmber of iterations or ntill [υ+] [υ] ɛ... PDE evoltion Te mes redistribtion is based on an iteration procedre and te PDE evoltion is independent of te first part. It can be solved by sing any ig resoltion finite volme metods. In tis paper, a second-order finite volme sceme wit a nmerical fl is te La-Friedrics fl: (.8) ˆf(a, b) = [f + f ma { f }(b a)]. Te details can be fond in Tang and Tang [9].

4 566 Syeon Sin and Woonjae Hwang 3. Nmerical eperiments In tis section, we consider te one-dimensional yperbolic conservation laws (3.) t + f() =, t >. As a scalar model, we consider Bckley - Leverett eqation. For a system model, we consider te sallow water eqation [5, 8]. We consider te Riemann problem for bot eqations. Some details are te following: te nmber of Jacobi iterations is 5; te sceme for evolving tis eqation is a (formally) second-order MUSCL finite volme sceme (wit te La-Friedrics fl) togeter wit a second-order Rnge-Ktta discretization; te CFL nmber sed is.3. Eample 3.. We consider Bckley-Leverett problem [8], (3.) t + f() =, f() = Te initial data are (, ) = + a( ), a =. { if < if >. Tis may be sed to model gas and oil in a reservoir, were is te flid satration(water). Tis fl is nonconve wit a single inflection point. Te caracteristic speed is f () = a( ). By following caracteristics, we can constrct te triple-valed soltion. By [ +a( ) ] te - - Figre. Nmerical soltions by MUSCL, Nmerical soltions by MUSCL wit MMM (J=5).

5 Te nmerical soltion by moving mes metods 567 eqal-area rle, tis triple-valed soltion replaced a sock. Ts Riemann soltion involves bot a sock and a rarefaction wave and is called a compond wave (a ) (b ) (c ) Figre.,, and sow nmerical soltions by MUSCL wit MMM of monitor parameter,, and, respectively. (a ),(b ), and (c ) sow mes trajectories of monitor parameter,, and, respectively. (J=5)

6 568 Syeon Sin and Woonjae Hwang Figre and sow te nmerical soltions at t = for niform grid and nonniform grid, respectively obtained wit te nmber of grids J = 5. Te monitor fnction sed in te comptation is (a ) (b ) (c ) Figre 3. Heigt:,, and sow nmerical soltions by MUSCL of J = 5, 75, and, respectively. (a ),(b ), and (c ) sow nmerical soltions by MUSCL wit MMM of J = 5, 75, and, respectively.

7 Te nmerical soltion by moving mes metods 569 w = + ξ. In tis eample, te monitor fnction is taken as w = + β ξ, β > were several vales of β are sed. In Figre, te nmerical soltion and mes trajectories wit tree different (a ) (b ) (c ) Figre 4. Velocity:,, and sow nmerical soltions by MUSCL of J = 5, 75, and, respectively. (a ),(b ), and (c ) sow nmerical soltions by MUSCL wit MMM of J = 5, 75, and, respectively.

8 57 Syeon Sin and Woonjae Hwang monitor fnctions(β =,, ) are plotted. It is observed tat te grid distribtions seem more reasonable, wit more points were te vale of crvatre is large. However, we avoid clstering too many points in te neigborood of te discontinities. Ts parameter β = of monitor fnction is efficient coice. Obviosly, te stdy on ow to coose te monitor constants seems very sefl. In principle, te monitor fnction w can be any appropriately cosen measre of te nmerical error in te soltion of te PDE. At points were te error is large, w sold also be large so tat mes points will tend to concentrate in tose areas were iger resoltion is needed Figre 5.,, and sow mes trajectories for J = 5, 75, and, respectively.

9 Te nmerical soltion by moving mes metods 57 Eample 3.. We apply moving mes algoritm to te one-dimensional sallow water eqations, ( ) ( ) + + = g t (a ) (b ) (c ) Figre 6. Heigt:,, and sow nmerical soltions by MUSCL of J =, 4, and 8, respectively. (a ),(b ), and (c ) sow nmerical soltions by MUSCL wit MMM of J =, 4, and 8, respectively.

10 57 Syeon Sin and Woonjae Hwang were, and g are eigt, velocity and gravitational constant, respectively. Consider te sallow water eqation wit te piecewise-constant (a ) (b ) (c ) Figre 7. Velocity:,, and sow nmerical soltions by MUSCL of J =, 4, and 8, respectively. (a ),(b ), and (c ) sow nmerical soltions by MUSCL wit MMM of J =, 4, and 8, respectively.

11 Te nmerical soltion by moving mes metods 573 initial data (, ) = { 3 if < if >, (, ) =. Tis is special case of te Riemann problem in wic l = r =, and is called te dam-break problem becase it models wat appens if a dam separating two levels of water brsts at time t =. Figres 3 and 4 sow te soltions of eigt and velocity, respectively at t = for niform and nonniform grid, obtained wit J = 5, Figre 8.,, and sow mes trajectories for J = 5, 75, and, respectively.

12 574 Syeon Sin and Woonjae Hwang (a ) (b ) (c ) Figre 9. Heigt:,, and sow nmerical soltions by MUSCL wit MMM of monitor parameter,, and respectively. (a ),(b ), and (c ) sow mes trajectories of monitor parameter,, and, respectively. (J=4)

13 Te nmerical soltion by moving mes metods 575 J = 75, and J =. Figre 5 sows te trajectories of te grid points at t=, obtained wit J=5, J=75, and J=. In tis eample, te monitor fnction sed in te comptation is w = + ( ξ ma ξ ξ ). Eample 3.3. Consider te sallow water eqation wit te piecewiseconstant initial data (, ) =, { if < (, ) = if >. Te soltion is symmetric in wit (, t) = (, t) and (, t) = (, t) at all times. A sock waves moves in direction, bringing te flid to rest, since te middle state mst ave m = by symmetry. Figres 6 and 7 sow te soltions of eigt and velocity, respectively at t = for niform and nonniform grid, obtained wit J =, J = 4, and J = 8. Figre 8 sows te trajectories of te grid points at t=, obtained wit J=, J=4, and J=8. In tis eample, te monitor fnction sed in te comptation is w = + ( ξ ma ξ ξ ). In tis eample, te monitor fnction is taken as w = + β( ξ ma ξ ξ ), β > were several vales of β are sed. In Figre 9, te mes trajectories and nmerical soltion wit tree different monitor fnctions(β =,, ) are plotted. References [] B. N. Azarenok, S. A. Ivanenko, and T. Tang, Adaptive mes redistribtion metod based on Godnov s sceme, Commnications in Matematical Sciences (3), [] B. N. Azarenok and T. Tang, Second-order Godnov-type sceme for reactive flow calclations on moving meses, Jornal of Comptational Pysics 6 (5), [3] Y. Di, R. Li, T. Tang and P.W. Zang, Moving mes finite element metods for te incompressible Navier-Stokes eqations, AMJ. Sci. Compt. 6 (5), [4] A. S. Dvinsky, Adaptive grid generation from armonic maps on Riemannian manifolds, Jornal of Comptational Pysics, 95 (99), [5] J. Felcman and L. Kadrnka, On a Moving Mes Metod Applied to te Sallow Water Eqations, Nmerical Analysis and Applied Matematics, International Conference (), [6] A. Harten, J. M. Hyman, Self-adjsting grid metods for one-dimensional yperbolic conservation laws, Jornal of Comptational Pysics 5 (983),

14 576 Syeon Sin and Woonjae Hwang [7] W. Hang, Y. Ren and R. D. Rssell, Moving mes metods based on moving mes partial differential eqations, Jornal of Comptational Pysics 3 (994), [8] R. J. Leveqe, Finite volme metods for yperbolic problems, Cambridge University Press, Cambridge,. [9] R. Li, T. Tang and P. Zang, Moving mes metods in mltiple dimensions based on armonic maps, Jornal of Comptational Pysics 7 (), [] R. Li, T. Tang and P. Zang, A moving mes finite element algoritm for singlar problems in two and tree space dimensions, Jornal of Comptational Pysics 77 (), [] S. Li, L. Petzold, Moving mes metods wit pwinding scemes for timedependent PDEs, Jornal of Comptational Pysics 3 (997), [] Y. Ren, R. Ren and Rssell, Moving mes partial differential eqations(mmpdes) based on te eqidistribtion principle, SIAM Jornal on Nmerical Analysis 3 (994), [3] K. Saleri, S. Steinberg, Fl-corrected transport in a moving grid, Jornal of Comptational Pysics (994), 4-3. [4] S. Sin, Adaptive mes metod for conservation laws, Master tesis, Korea University, Seol, 6. [5] S. Sin and W. Hwang, Nmerical soltion of brgers eqation by moving mes metods, Proceedings of te National Institte for Matematical Sciences (6), -3. [6] J. M. Stockie, J. A. Mackenzie, and R. D. Rssell, A moving mes metod for one-dimensional yperbolic conservation laws, SIAM Jornal on Scientific Compting (), [7] Z. Tan, Z. Zang, Y. Hang, and T. Tang, Moving mes metods wit locally varying time steps, Jornal of Comptational Pysics (4), [8] H. Tang, Soltion of te sallow-water eqations sing an adaptive moving mes metod, International Jornal for Nmerical Metods in Flids 44 (4), [9] H. Tang and T. Tang, Adaptive mes metods for one-and two-dimensional yperbolic conservation laws, SIAM Jornal on Nmerical Analysis 4 (3), [] H. Z. Tang and T. Tang, Mlti-dimensional moving mes metods for sock comptations, Inter. Conf. on Sci. Compt. and PDEs,on te Occasion of S. Oser s 6t birtday, Dec. -5,, Hongkong. Contemporary Matematics 33 (3), [] H. Z. Tang, T. Tang, P. Zang, An adaptive mes redistribtion metod for nonlinear Hamiton-Jacobi eqations in two- and tree-dimensions, Jornal of Comptational Pysics 88 (3), [] T. Tang, Moving mes metods for comptational flid dynamics, Contemporary Matematics 383 (5), [3] P. A. Zegeling, W. D. de Boer, and H. Z. Tang, Robst and efficient adaptive moving mes soltion of te -D Eler eqations, Contemporary Matematics 383 (5),

15 Te nmerical soltion by moving mes metods 577 [4] Z. Zang and T. Tang, An adaptive mes redistribtion algoritm for convection-dominated problems, Commnications on pre and Applied Analysis (), [5] A. Winslow, Nmerical soltion of te qasi-linear Poisson eqation in a nonniform triangle mes, Jornal of Comptational Pysics (967), * Department of Matematics Korea University Seol 36-7, Repblic of Korea angelic5@korea.ac.kr ** Department of Information and Matematics Korea University Jociwon 339-7, Repblic of Korea woonjae@korea.ac.kr