Copyright information to be inserted by the Publishers. Unsplitting BGK-type Schemes for the Shallow. Water Equations KUN XU

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1 Copyright information to be inserted by the Pubishers Unspitting BGK-type Schemes for the Shaow Water Equations KUN XU Mathematics Department, Hong Kong University of Science and Technoogy, Cear Water Bay, Kowoon, Hong Kong As a continuation of the research on the BGK type schemes, in this paper we present a second order unspitting method for the shaow water equations. Dierent from any other approaches, the source terms are incuded expicity into the time-dependent ux functions across ce interface. Introduction The deveopment of unspitting numerica schemes for the hyperboic equations with source terms has obtained much attention in recent years. Based argey on the macroscopic equations and wave modeing, it is dicut to deveop such kind of schemes. In this paper, for the rst time, an unspitting gas-kinetic scheme for the shaow water equations is deveoped. The shaow water equations in -D case can be written as W t + F (W ) x = S(W ); () where W is the vector of ow variabes and F (W ) the corresponding uxes. The source term S(W ) accounts for dynamica eects on the ow motion from the wavy river bottom. If we integrate the above equation in a numerica ce j from x j?= to x j+=, and in a time step from t n to t n+, Z t n+ Z xj+= t n x j?= (W t + F (W ) x )dxdt = Z t n+ Z xj+= t n x j?= S(W )dxdt; we can get W n+ j?wj n = Z t n+? Fj?= (t)? F j+= (t) dt+ Z xj+= Z t n+ S(W )dxdt; x t n x x j?= t n () where x = x j+=? x j?= and W j is the average vaue of W in ce j, i.e. R W j = xj+= x x j?= W dx. The above equation is an exact formuation for a nitevoume scheme.

2 K. XU In order to update W in Eq.(), a spitting technique is usuay used, where W t + F (W ) x = (3) is soved to evauate the numerica uxes and S(W ) is treated impicity or expicity inside each ce. Or, these two stages are couped using Runge-Kutta methods. The spitting schemes treat the ow around ce interface and inside each ce dierenty. Eq.(3) totay ignores the source term eects on the ow motion around ce interface. Physicay, the source term never knows where the ce interface is, it denitey eects the ow motion everywhere in space and time. Therefore, the spitting scheme destroys the intrinsic homogeneous of physica space and this distortion can be never eiminated from Runge-Kutta techniques. As a continuation of previous research 4,5, the focus of this paper is to deveop an unspitting BGK-type scheme for the shaow water equations (Eq.()), where the numerica uxes and source terms are couped and soved simutaneousy in both space and time. Unspitting BGK Scheme for the Shaow Water Equations We consider the shaow water equations with source term: U + U U + = x G t?h (x)g : (4) In this system, is the water height, U the veocity, and G the gravitationa constant. H(x) is the shape function of river bottom. The generaized BGK mode for the above equation can be constructed as f t + uf x = g? f + s; (5) where f is the gas distribution function and g the equiibrium state f approaches. is the partice coision time and s the additiona source term. For the shaow water equations, a f, g and s are functions of space x, time t, and partice veocity u. The equiibrium state g is a Maxweian g = e n?(u?u) = ( ) e?(u?u) ; where = ( ) and is reated to and G with the reation = G. Due to the conservation property in partice coisions, f and g satisfy the compatibiity condition Z (f? g) du = for = (; u) T (6) at any points in space and time. The corresponding source term s in Eq.(5) can be constructed as s = s? s, where s = e ) = ( ) )

3 and Unspitting BGK-type Schemes for the Shaow Water Equations 3 s = e ) = ( ) ) : The numerica uxes for the rst term on the right hand side of Eq.() are based on the integra soution of the BGK mode (Eq.(5)), f(x j+= ; t; u) = Z t (g(x ; t ; u) + s(x ; t ; u)) e?(t?t )= dt +e?t= f (x j+=? ut); (7) where x j+= is the ce interface and x = x j+=? u(t? t ) the partice trajectory. There are three unknowns in the above equation. One is the initia gas distribution function f at time t =. The others are g and s in both space and time ocay around (x j+= ; t = ). The numerica scheme for soving Eq.(7), aong with the compatibiity condition(6), is described as foows: Step(): Use MUSCL technique to interpoate the conservative variabes 3 W j = ( j ; j U j ) in each ce, and get the reconstructed initia data W j (x) = W j (x j )+ W j(x j+= )? W j (x j?= ) x j+=? x j?= (x?x j ) for x[x j?= ; x j+= ]: (8) Step(): Based on the reconstructed data in Step(), around each ce interface x j+=, construct the initia gas distribution function f,? g + a f (x) = (x? x j+= ) ; x x j+= g? r + a r (x? x j+= ) ; x x j+= ; (9) where the states g and g r are the Maxweian distribution functions which have the one-to-one correspondence to the conservative variabes at ce interface; g = g (W j (x j+= )) and g r = g r (W j+ (x j+= )): () For exampe, if we assume g = e n? (u?u ) a coecients in g can be obtained j (x j+= ) U A j U j (x j+= )= j (x j+= ) =(G ) = ( ) e? (u?u ) ; () Simiar equations can be found for g r. a ;r in Eq.(9) have the forms a ;r = m ;r + m;r u + m;r 3 u : A : ()

4 4 K. XU The coecients (m ;r ; m;r ; m;r 3 ) can be determined from ( ; U ) and ( r ; r U r ) and the sopes of the reconstructed mass and momentum in @ The reations are m ;r 3U = ( + U m ;r =? + ;r m ;r 3 : (3) So, a coecients in Eq.(9) have been obtained from the initiay reconstructed data in Step(). For simpication, we wi assume x j+= = in the rest of this paper. Step(3): The equiibrium state g is assumed to be continuous across a ce interface g = g? + (? H[x])a x + H[x]a r x + At ; (4) where H(x) is the heaviside function and g the state ocated at (x = ; t = ), g = e n?(u?u) = ( ) e? (u?u ) : (5) a ; a r, and A in Eq.(4) have the forms a ;r = m ;r + m;r u + m;r 3 u ; A = A + A u + A3 u : (6) Taking both imits (x! ; t! ) in Eq.(7) and (4), and appying the compatibiity condition (Eq.(6)) at (x = ; t = ), the macroscopic variabes reated to g can be uniquey determined in terms of f, Z Z W = = g U du = [g H(u) + g r (? H(u))] du; (7) where W is the \averaged " ow variabes at the ce interface. Then, connecting W to the ce centered vaues W j (x j ) and W j+ (x j+ ), we can get two sopes for mass and U ) T = W? W j (x j ) for x x j+=? x U ) r T = W j+(x j+ )? W for x ; x j+? x j+=

5 Unspitting BGK-type Schemes for the Shaow Water Equations 5 from which (a ; a r ) can be determined in a simiar way as shown in Eq.(3). Now, the ony unknown in Eq.(4) is A, which can be expressed as A = + G? ; G ; A =? G G A 3 = G are unknowns at this time. Step(4): The source term (s = s? s ) can be constructed as? s = s + (? H[x])a s x + H[x]a r sx + As t where a ;r s and a;r ; (9) s = s? + (? H[x])a s x + H[x]a r sx + As t ; () s = ( ) e? ) ; s = ( ) e? (u? in Eq.() can be written as a ;r s =!;r +!;r u +!;r 3 u ) : () a ;r s = ;r + ;r u + ;r 3 u : The coecients in the above equations can be easiy obtained in terms of ( ; U ) and their sopes in Eq.(8). For exampe, for s, we have and for s,! ;r = + @ G ;r! ;r 3 = @ ;r ;r =! ;r ; ;r =?! ;r ; ;r 3 =! ;r 3 : So, a parameters in Eq.() at t = have been determined. The unknown terms in () are those reated to the time derivative of the source terms, A s = + u + 3 u

6 6 K. XU and which are reated = A s = + u + 3 u ; + ) @x 3 G : = =? 3 = 3 : Step(5): Substitute Eq.(4),() and (9) into the integra soution (7), we can obtain the distribution function f at x =, f(; t; u) = (g + (s? s )) where? + a H[u] + a r (? H[u]) ug +? (a H[u] + s ar ( s? H[u]))us? (a H[u] + s ar ( s? H[u]))us? + Ag + ( A s s? A s s ) + 3? (? uta )H[u]g + (? uta r )(? H[u])g r ; () =? e?t= ; = (? + e?t= ) + te?t= ; = (t=? + e?t= ); 3 = e?t= : As pointed out in the ast two sections, there are two unknowns ) in Eq.(). Since the compatibiity condition must be satised everywhere in space and time, it can be integrated in a whoe CFL time step T at x = Z T Z (f(; t; u)? g(; t; u)) dtdu = ; (3) from (? 5 5?? Z )T = g +? u? a H[u] + a r (? H[u]) g? +? 3 H[u]g + (? H[u])g r +? 4 u? a H[u]g + a r (? H[u])g r du (4)

7 Unspitting BGK-type Schemes for the Shaow Water Equations 7 is obtained. A terms on the right hand side of the above equation are known and = T? (? e?t = );? =?T + (? e?t = )? T e?t = ;? = T? T + (? e?t = );? 3 = (? e?t = );? 4 =?T e?t = + (? e?t = ) ;? 5 = T? (? e?t = ) : So, ) in Eq.(4) can be obtained easiy. Step(6): The time-dependent numerica uxes of mass and momentum across the ce interface can be obtained using Eq.(), Z F (t) = u F U (t) f j+= (; t; u)du; (5) j+= from which the time variation of mass density inside each ce j can be evauated Z t Z j (t) = j () + u? f j?= (; t; u)? f j+= (; t; u) dtdu: Since j (t) is a function of time, the integration of the source term?h (x)g in a whoe time step can be found anayticay?h (x)g Z T j (t)dt: So, to the second-order of accuracy, both terms on the right hand side of Eq.() are obtained from the BGK scheme. 3 Numerica Exampes Two numerica exampes for the shaow water equations are presented in this section. The computationa domain in both cases is x[; ]. The MUSCL imiter is used for interpoations of and U in each ce at the beginning of each time step. The time step T is determined using CFL=:65. The coision time is a oca constant = C T p + C T j? r r j + r r ; (6) where C = :; C = : are xed in a cacuations. The update of ow variabes is based on Eq.(), where the time-dependent ux function F j+= (t) and the expicit time integration S(W j (t)) are given at the end of ast section.

8 8 K. XU.8 Density Momentum x FIGURE : Shock Tube Test Case with Points for Shaow Water Equations In order to vaidate the scheme, the rst test case is a standard shock tube test without source term. The initia condition for this case is ( = :; U = :)j x<:5 and ( r = :5; U r = :)j x:5 : The simuation resuts with grid points at time T = :3 are shown in Fig.(), where the soid ine is the exact soution. From this test case, we can observe that the BGK scheme gives accurate resuts in both discontinuous and smooth regions. The second test case is about to study the convergence property of the unspitting scheme and compare that with the resuts from spitting scheme. The spitting scheme is dierent from the unspitting scheme ony in the evauation of the numerica uxes, where the homogeneous hyperboic equations without the source terms (Eq.(3)) are soved. The method for soving the homogeneous equation is aso based on the BGK mode f t + uf x = (g? f)=: (7) The numerica procedures for soving Eq.(7) is the same as that presented in the ast section except deeting a terms reated to s in step(4) and step(5). As a specia case, we are going to test the accuracy of both schemes to keep an exact soution for the shaow water equations and study the convergence property. The shape function of the river bottom is described by the exponentia function H(x) = :e?(x?:5) L where L = :. The initia condition is an exact steady state soution of Eq.(4),

9 Unspitting BGK-type Schemes for the Shaow Water Equations 9 where the ow variabes are =? H(x) and U = : In order to keep the above exact soutions (W n+ j = Wj n ), the two terms on the right hand sides of Eq.() shoud compensate each other. However, for the spitting scheme, the ux function F j+= (t) does not account for the existence of source term. So, it seems impossibe for these two terms to cance each other accuratey. On the other hand, the unspitting scheme incudes source term eects in the ux function, to the second order accuracy both terms shoud certainy cance each other partiay. Therefore, the unspitting shoud have a more accurate soution than that from the spitting scheme. In this case, we have used 7 dierent ce sizes x as shown in Tabe() to study the convergence property in both unspitting and spitting schemes. The error indicator L -norm is dened as P N j= L = j j(t) + H(x)? j ; N where N is the tota number of ces N = x. Athough the initia input data is an exact soution, the numerica schemes can not keep this soution. Any schemes wi generate articia waves from wavy bottom region and these errors coud propagate throughout the whoe computationa domain and become background noise as the computationa time is ong enough. In order to observe both the initia transient and asymptotic behavior of the numerica soution, we have measured the L error at two dierent output time for both schemes. Some of the computationa resuts are printed in Tabe(). Fig.() is a pot of the resuts in a ogarithmic scae for the tota mesh points N and L error. As indicated from this Figure, the unspitting scheme amost have consistent errors at both T = : and T = : for dierent tota mesh points, which means that the unspitting scheme has no obvious initia transient period and it converges to its na asymptotic soution quicky. On the other hand, the spitting scheme takes much onger time to baance both terms on the right hand sides of Eq.(). It converges sowy to the exact soution, especiay at eary time T = :. In order to get the same accuracy, the spitting scheme needs more than times grid points as that used in unspitting scheme. 4 Concusion In this paper, we have deveoped the BGK scheme for the shaow water equations with source term. The source term eect are expicity incuded in the timeevoution of the gas distribution at a ce interface, from which the numerica uxes are obtained. The current approach, once again, demonstrates the generaity, robustness, and accuracy of the BGK type schemes in soving hyperboic conservation aws. The scheme presented in this paper, as far as we know, is the rst truey unspitting scheme for the shaow water equations.

10 K. XU TABLE : Mesh Renement Study for Shaow Water Equations Unspitting Scheme Spitting Scheme x L error at T=. L error at T=. L error at T=. L error at T=..453e-4 3.8e e-4.799e-3.495e-5 4.7e e-5.46e e-6.897e e e e e e e e e-6.683e e-4.579e-7.55e e-6.63e e e e e e e e e e-8.38e e e e e-9.46e e e-.393e e-8.96e e e- 5.35e e-6 5 Grid Refinement Study of Spitting and Unspitting Schemes 3 4 og(l_ Error) T=. (Spitting) * T=. (Unspitting) o T=. (Spitting) x T=. (Unspitting) og(n) FIGURE : Mesh Renement Study of Unspitting and Spitting Schemes for Shaow Water Equations

11 REFERENCES Unspitting BGK-type Schemes for the Shaow Water Equations. S. Jin, \Runge-Kutta Methods for Hyperboic Conservation Laws with Sti Reaxation Terms", J. Comput. Phys.,, 5-67, R.B. Pember, \Numerica Methods for Hyperboic Conservation Laws with Sti Reaxation, II. Higher-Order Godunov Methods", SIAM J. Sci. Comput. 4, No.4, , B. van Leer, \Towards the Utimate Conservative Dierence Scheme V. A Second Order Seque to Godunov's Method", J. of Comput. Phys. 3, 76-99, K. Xu, C. Kim, L. Martinei,and A. Jameson, \BGK-Based Schemes for the Simuation of Compressibe Fow" to appear in Int. J. of Comput. Fu. Dyn., K. Xu, \A Gas-Kinetic Scheme for Hyperboic Conservation Laws with Source Terms", submitted to J. Comput. Phys., Juy 996.