DSpace at VNU: Properties of the Wave Curves in the Shallow Water Equations with Discontinuous Topography

Bull Malays Math Sci Soc (2016) 39:305–337 DOI 10.1007/s40840-015-0186-1 Properties of the Wave Curves in the Shallow Water Equations with Discontinuous Topography Mai Duc Thanh1 · Dao Huy Cuong2 Received: 23 April 2013 / Revised: October 2013 / Published online: August 2015 © Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015 Abstract We first establish the monotonicity of the curves of composite waves for shallow water equations with discontinuous topography Second, a critical investigation of the Riemann problem yields deterministic results for large data on the existence of Riemann solutions made of Lax shocks, rarefaction waves, and admissible stationary contacts Although multiple solutions can be constructed for certain Riemann data, we can determine relatively large neighborhoods of Riemann data in which the Riemann problem admits a unique solution Keywords Shallow water equations · Discontinuous topography · Shock wave · Nonconservative · Composite wave · Monotonicity · Riemann problem Mathematics Subject Classification 76L05 Primary 35L65 · 74XX · Secondary 76N10 · Introduction The curves of composite waves in the shallow water equations with discontinuous topography play an essential role in solving theRiemann problem However, the Communicated by Ahmad Izani MD Ismail B Mai Duc Thanh mdthanh@hcmiu.edu.vn Dao Huy Cuong cuongnhc82@gmail.com Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam Nguyen Huu Cau High School, 07 Nguyen Anh Thu, Trung Chanh, Hoc Mon, Ho Chi Minh City, Vietnam 123 306 M D Thanh, D H Cuong monotonicity of this kind of curves has not been proved, partly due to its complicatedness of composing waves in one characteristic family along a curve of waves in another characteristic family This gives us a motivation for this study Furthermore, we provide in this paper a critical investigation the Riemann problem for shallow water equations with discontinuous topography Precisely, the model is given by ∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x h u + gh = −gh∂x a, ∂t a = 0, (1.1) where the height of the water from the bottom to the surface, denoted by h, and the fluid velocity u are the main unknowns Here, g is the gravity constant, and a = a(x) (with x ∈ R) is the height of the bottom from a given level Observe that the third equation in (1.1) is a trivial equation The Riemann problem for (1.1) is the Cauchy problem with the initial data, called the Riemann data, of the form (h, u, a)(x, 0) = (h L , u L , a L ), for x < 0, (h R , u R , a R ), for x > (1.2) It has been known that the system of balance laws in nonconservative form (1.1) is hyperbolic whose characteristic fields may coincide, see [15] for example Building Riemann solutions of this kind of systems would involve the construction of curves of composite waves, which include waves in a genuinely nonlinear and a linearly degenerate characteristic fields For example, in the general case a (local) existence result was established by Goatin and LeFloch [5] The Riemann problem for (1.1) was studied in Refs [15,16], where the existence for large data was obtained Furthermore, it was shown in Ref [16] that up to three solutions can be constructed for certain Riemann data However, the monotonicity property of the curves of composite waves has not been proved before That leaves an open question for the completeness of the theory of this kind of systems Furthermore, when does a Riemann solution existence, precisely? Hence, the current paper has two goals: the first goal is to establish the monotonicity property of the curves of composite waves—and so the domain of uniqueness could be found, and the second goal is to seek for a deterministic version of the existence of Riemann solutions—where explicit large domains of existence could be found Systems of balance laws in nonconservative form have attracted many authors A general framework for systems of balance laws in nonconservative form was introduced by Dal Maso-LeFloch-Murat [4], see also LeFloch [13] The standard admissibility criterion for shock waves for hyperbolic systems of conservation laws was addressed in the pioneering work by Lax [12] Shock waves and the related traveling waves in scalar conservation laws with a nonzero right-hand side were studied by Isaacson-Temple [7,8] and Thanh [24] As mentioned above, a local existence of Riemann solutions for general systems of balance laws with resonance was established by Goatin-LeFloch [5] The Riemann problem for fluid flows in a nozzle with discontinuous cross-section were considered by MarchesinPaes-Leme 123 Properties of the Wave Curves in the Shallow Water 307 [17], Andrianov-Warnecke [1], LeFloch-Thanh [14], Kroener-LeFloch-Thanh [11], and Thanh [20] Recently, the Riemann problem and exact solutions for two-phase flow models are considered by Andrianov-Warnecke [2], Schwendeman-Wahle-Kapila [19], and Thanh [22,23] Numerical schemes for shallow water equations were studied by Chinnayya-LeRoux-Seguin [3], Thanh-Fazlul-Ismail [16,21], Jin-Wen [9,10], Rosatti-Begnudelli [18], and Gallardo-Parés-Castro [6] See also the references therein The organization of this paper is as follows In Sect we recall basic concepts and properties of the system (1.1) In Sect we study the monotonicity of the curves of composite waves Finally, in Sect we present deterministic version of existence results, and uniqueness of Riemann solutions of the problems (1.1)–(1.2) Preliminaries 2.1 Wave Curves The system (1.1) can be re-written as a nonconservative system as ∂t U + A(U )∂x U = 0, (2.1) where ⎛ ⎞ h U = ⎝u ⎠ , a ⎛ u A(U ) = ⎝g h u ⎞ g⎠ The matrix A = A(U ) has three real eigenvalues λ1 (U ) := u − gh < λ2 (U ) := u + gh, λ3 (U ) := 0, (2.2) together with the corresponding eigenvectors which can be chosen as ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ gh h √ √h r1 (U ) := ⎝− gh ⎠ , r2 (U ) := ⎝ gh ⎠ , r3 (U ) := ⎝ −gu ⎠ 0 u − gh (2.3) Thus, the system (2.1) is hyperbolic Moreover, the first and the third characteristic speeds can coincide, i.e., λ1 (U ) = λ3 (U ) = 0, on the surface C+ := (h, u, a)| u = gh , (2.4) and the second and the third characteristic fields can coincide, i.e., λ2 (U ) = λ3 (U ) = 0, 123 308 M D Thanh, D H Cuong on the surface C− := (h, u, a)| u = − gh (2.5) The above argument means that the system (1.1)–(1.2) is hyperbolic, but is not strictly hyperbolic Besides, it is easy to see that the first and second characteristic fields (λ1 , r1 ), (λ2 , r2 ) are genuinely nonlinear, that is ∇λ1 · r1 = 0, ∇λ2 · r2 = 0, and that the third characteristic field (λ3 , r3 ) is linearly degenerate, that is ∇λ3 · r3 = Set C := C+ ∪ C− , G := U | λ1 (U ) > λ3 (U ) = U | u > gh , G := U | λ2 (U ) > λ3 (U ) > λ1 (U ) = U | |u| < G+ := {U ∈ G | u ≥ 0} = U | ≤ u < gh , gh , G− := {U ∈ G | u < 0} = U | > u > − gh , G := U | λ3 (U ) > λ2 (U ) = U | u < − gh (2.6) As discussed in [15], across a discontinuity there are two possibilities: (i) either the bottom height a remains constant, (ii) or the discontinuity is stationary (i.e., propagates with zero speed) Let us consider the first case (i), where the system (1.1) is reduced to the usual shallow water equations with flat bottom Then, we can determine the Rankine–Hugoniot relations and the admissibility criterion for shock waves as usual Let us recall that a shock wave of (1.1) is a weak solution of the form U (x, t) = U− , x < st, U+ , x > st, (2.7) where U− , U+ are the left-hand and right-hand states, respectively, and s = s(U− , U+ ) is the shock speed A shock wave (2.7) is admissible, called an i-Lax shock, if it satisfies the Lax shock inequalities, see [12], λi (U+ ) < s(U− , U+ ) < λi (U− ), i = 1, From now on, we consider admissible shock waves, only 123 (2.8) Properties of the Wave Curves in the Shallow Water 309 Given a left-hand state U0 , the set of all right-hand states that can be connected to U0 by an i-Lax shock forms a curve, denoted by Si (U0 ), i = 1, In a backward way, given a right-hand state U0 , the set of all left-hand states that can be connected to U0 by an i-Lax shock forms a curve, denoted by SiB (U0 ), i = 1, These curves are defined by S1 (U0 ) : u = u0 − g (h − h ) + , h > h0, h h0 S2 (U0 ) : u = u0 + g (h − h ) + , h < h0, h h0 S1B (U0 ) : u = u0 − g (h − h ) + , h < h0, h h0 S2B (U0 ) : u = u0 + g (h − h ) + , h > h0, h h0 (2.9) see [15] It is interesting that the shock speeds in the nonlinear characteristic fields may coincide with the characteristic speed of the linearly degenerate field as stated in the following lemma Lemma 2.1 (Lem 2.1, [16]) Consider the projection on the (h, u)-plan To every U L = (h L , u L ) ∈ G there exists exactly one point U L# ∈ S1 (U L ) ∩ G + such that the 1-shock speed λ1 (U L , U L# ) = The state U L# = (h #L , u #L ) is defined by h #L = −h L + h 2L + 8h L u 2L /g , u #L = uLhL h #L Moreover, for any U ∈ S1 (U L ), the shock speed λ1 (U L , U ) > if and only if U is located above U L# on S1 (U L ) Next, let us consider rarefaction waves, which are piecewise smooth self-similar solutions of (2.1) It was shown by [?] that the bottom height a remains constant through any rarefaction fan Given a left-hand state U0 , the set of all right-hand states that can be connected to U0 by an i-rarefaction waves of (2.1) forms a curve, denoted by Ri (U0 ), i = 1, In a backward way, given a right-hand state U0 , the set of all left-hand states that can be connected to U0 by an i-rarefaction wave forms a curve, denoted by RiB (U0 ), i = 1, These curves are given by √ √ u = u0 − g h − h0 , h ≤ h0, √ √ R2 (U0 ) : u = u + g h − h0 , h ≥ h0, √ √ R1B (U0 ) : u = u − g h − h0 , h ≥ h0, R1 (U0 ) : 123 310 M D Thanh, D H Cuong R2B (U0 ) : √ √ u = u0 + g h− h0 , h ≤ h0, (2.10) see [15] We can therefore define the forward and backward wave curves in the nonlinear characteristic fields as follows: Wi (U0 ) = Ri (U0 ) ∪ Si (U0 ), WiB (U0 ) = RiB (U0 ) ∪ SiB (U0 ), i = 1, (2.11) As seen above, the curves Wi (U0 ) can be parameterized as a function u = wi (U0 ; h) of h ≥ 0, and the curves WiB (U0 ) can be parameterized as a function u = wiB (U0 ; h) of h ≥ 0, i = 1, Precisely, W1 (U0 ) : u = w1 (U0 ; h) := W2 (U0 ) : u = w2 (U0 ; h) := W1B (U0 ) : W2B (U0 ) : ⎧ √ √ ⎨ u − 2√ g h − h0 , h ≤ h0, ⎩ u − g (h − h ) + , h > h , h h0 ⎧ √ √ √ ⎨ u0 + g h − h0 , h ≥ h0, ⎩ u + g (h − h ) + , h < h , h h0 ⎧ √ √ √ ⎨ u0 − g h − h0 , h ≥ h0, u = w1B (U0 ; h) =: ⎩ u − g (h − h ) + , h < h , h h0 ⎧ √ √ √ ⎨ u0 + g h − h0 , h ≤ h0, u = w2B (U0 ; h) := (2.12) g 1 ⎩ u0 + (h − h ) h + h , h > h It was shown in [15] that w1 (U0 ; h) and w1B (U0 ; h) are strictly convex and strictly decreasing functions of h, while w2 (U0 ; h) and w2B (U0 ; h) are strictly concave and strictly increasing functions of h ≥ Let us now consider the case (ii), where the discontinuity satisfies the jump relations [hu] = 0, u2 + g(h + a) = The last jump relations determine the stationary-wave curve (parameterized with h) as follows: W3 (U0 ) : u = w3 (U0 ; h) := a = a0 + u0h0 , h ≥ 0, h u 20 − u + h − h 2g (2.13) It is easy to check that the function w3 (U0 ; h), h ≥ 0, is strictly convex and strictly decreasing for u > 0, and strictly concave and strictly increasing for u < 123 Properties of the Wave Curves in the Shallow Water 311 2.2 Properties of Stationary Contacts Given a state U0 = (h , u , a0 ) and another bottom level a = a0 , we let U = (h, u, a) be the corresponding right-hand state of the stationary contact issuing from the given left-hand state U0 We now determine h, u in terms of U0 , a, as follows Substituting u = h u / h from the first equation of (2.13) to the second equation of (2.13) , we obtain h0u0 a0 + (2.14) + h − h = a u 20 − 2g h Multiplying both sides of (2.14) by 2gh , and then re-arranging terms, we get F(h) = F(U0 , a; h) := 2gh + 2g(a − a0 − h ) − u 20 h + h 20 u 20 = (2.15) We easily check F(0) = h 20 u 20 ≥ 0, F (h) = 6gh + 2g(a − a0 − h ) − u 20 h, F (h) = 12gh + 2g(a − a0 − h ) − u 20 , so that F (h) = iff h = or h = h ∗ = h ∗ (U0 , a) := u 20 + 2g(a0 + h − a) 3g (2.16) u 20 , then h ∗ < and F (h) > 0, h > Since F(0) ≥ 0, there 2g u2 is no root for (2.15) Otherwise, if a ≤ a0 + h + , then F (h) > 0, h > h ∗ and 2g F (h) < 0, < h < h ∗ In this case, F(h) has two zeros h if and only if If a > a0 + h + Fmin := F(h ∗ ) = −gh 3∗ + h 20 u 20 ≤ 0, or h ∗ ≥ h (U0 ) := (h 20 u 20 /g)1/3 It is easy to check that h ∗ ≥ h (U0 ) if and only if a ≤ amax (U0 ) := a0 + h + u 20 − 1/3 (h u )2/3 2g 2g Lemma 2.2 (Lem 2.2, [16]) Given a state U0 = (h , u , a0 ) and a bottom level a = a0 The following conclusions hold 123 312 M D Thanh, D H Cuong (i) amax (U0 ) ≥ a0 , amax (U0 ) = a0 if and only if (h , u ) ∈ C± (ii) The nonlinear equation (2.14) admits a root if and only if a ≤ amax (U0 ), and in this case it has two roots ϕ1 (a) ≤ h ∗ (U0 , a) ≤ ϕ2 (a) Moreover, if a < amax (U0 ), these two roots are distinct (iii) According to the part (ii), whenever a ≤ amax (U0 ), there are two states Ui (a) = (ϕi (a), u i (a), a), where u i (a) = h u /ϕi (a), i = 1, to which a stationary contact from U0 is possible Moreover, the locations of these states can be determined as follows: U1 (a) ∈ G if u > 0, U1 (a) ∈ G if u < 0, U2 (a) ∈ G We next prove the following result, which has been stated in [15] without a proof Lemma 2.3 We have the following comparisons (i) If a < a0 , then ϕ1 (a) < h < ϕ2 (a) (ii) If a0 < a < amax (U0 ), then h < ϕ1 (a) for U0 ∈ G ∪ G , h > ϕ2 (a) for U0 ∈ G Proof We have F(h ) = 2g(a − a0 )h If a < a0 , then a < a0 ≤ amax (U0 ) and F(h ) < It implies that F(h) has two zeros ϕ1,2 (a) such that ϕ1 (a) < h < ϕ2 (a) If a0 < a < amax (U0 ), then F(h ) > and F(h) has two distinct zeros ϕ1,2 (a) such that h < ϕ1 (a) or h > ϕ2 (a) u 20 h 20 1/3 > h In the case g U0 ∈ G , h > ϕ2 (a) since F (h ) = 2h [gh − u + 2g(a − a0 )] > In the case U0 ∈ G ∪ G , h < ϕ1 (a) since h ∗ = From Lemma 2.2 , we can construct two-parameter wave sets The Riemann problem may therefore admit up to a one-parameter family of solutions To select a unique 123 Properties of the Wave Curves in the Shallow Water 313 solution, we impose an admissibility condition for stationary contacts, referred to as the Monotonicity Criterion and defined as follows (MC) Along any stationary curve W3 (U0 ), the bottom level a is monotone as a function of h The total variation of the bottom level component of any Riemann solution must not exceed |a L − a R |, where a L , a R are left-hand and right-hand bottom levels A similar criterion was used in [15] Lemma 2.4 The Monotonicity Criterion implies that any stationary shock does not cross the boundary of strict hyperbolicity, in other words (i) If U0 ∈ G ∪ G , then only the stationary contact based on the value ϕ1 (a) is allowed (ii) If U0 ∈ G , then only the stationary contact using ϕ2 (a) is allowed Monotone Property of Curves of Composite Waves Observe that by the transformation x → −x, u → −u, a left-hand (right-hand) state U = (h, u, a) in G − (in G ∪ C− ) will be transformed to the right-hand (lefthand, respectively) state V = (h, −u, a) in G + (in G ∪ C+ , respectively) Thus, the construction of wave curves, and therefore, the Riemann solutions for Riemann data around C− can be obtained from the one for Riemann data around C+ Thus, without loss of generality, in the sequel we consider only the case, where Riemann data are in G ∪C+ ∪ G + Moreover, the construction will be relied on the left-hand state U L (and hence the region of the right-hand states will follow) if a L > a R , and the construction will be relied on the right-hand state U R (and hence the region of the left-hand states will follow), otherwise Notations (i) Wk (Ui , U j ) (Sk (Ui , U j ), Rk (Ui , U j )) denotes the kth-wave (kth-shock, kthrarefaction wave, respectively) connecting the left-hand state Ui to the right-hand state U j , k = 1, 2, (ii) Wm (Ui , U j )⊕Wn (U j , Uk ) indicates that there is an mth-wave from the left-hand state Ui to the right-hand state U j , followed by an nth-wave from the left-hand state U j to the right-hand state Uk , m, n ∈ {1, 2, 3} (iii) We will sometimes write for simplicity in this section the curves defined by (2.1) as u = wi (h) instead of u = wi (U0 ; h), and u = wiB (h) instead of u = wiB (U0 ; h), i = 1, 2, when U0 is clear, if this does not any confusion (iv) U # denotes the state resulted by a shock wave from U with zero speed; U denotes the state resulted by an admissible stationary contact from U 3.1 Case : U L ∈ G ∪ C+ and a L > a R Let U± = (h ± , u ± ) stand for the states at which the wave W1 (U L ) intersects with the curves C± , respectively From U L (a L ) the Riemann solution can begin with a 123 314 M D Thanh, D H Cuong stationary contact wave to some state U L0 (a R ) ∈ G using ϕ1 (U L , a R ) There is one 0# state U L0# (a R ) ∈ W1 (U L0 ) ∩ G + such that λ1 (U L , U L ) = and λ1 (U L , U ) > for 0# h 0L < h < h 0# L , λ1 (U L , U ) < for h > h L So, the solution can continue by a 1-wave 0# from U L to state U such that ≤ h ≤ h L The set of these states U form the curve composite W3→1 (U L ) The composite curve W3→1 (U L ) is a part of the curve W1 (U L0 ) The curve W1 (U L0 ) intersects axis h = at the point I = 0, u up := u 0L + gh 0L I and U L0# are two endpoints of the composite curve W3→1 (U L ) From U L the Riemann solution can begin with 1-shock to state A ∈ S1 (U L ) ∩ G such that λ1 (U L , A) ≤ So, A is between U L# and U− The solution continue with a stationary contact wave from state A(a L ) to state A0 (a R ) ∈ G using ϕ2 (A, a R ) The set of such states A0 forms the curve composite W1→3 (U L ) U L#0 and U−0 are two endpoints of this curve, where U−0 = (h 0− , u 0− ) = (ϕ2 (U− , a R ), w3 (U− , ϕ2 (U− , a R ))) Of course, the curve of composite waves can be constructed beyond h > h 0− However, in the region G , the characteristic speed λ2 (U ) < = λ3 (U ), U ∈ G In this case, the construction of the Riemann solution(as seen below) may not be well defined That is the reason why we stop the composite curve at U−0 Besides, at each level a ∈ [a R , a L ], from U L (a L ) the Riemann solution can begin with the stationary contact wave to some state B(a) ∈ G ∪ C+ using ϕ1 (U L , a) The solution is continued with 1-shock from state B to state B # ∈ G + ∪ C+ such that # λ1 (B, B ) = Then, the solution is continued with the stationary contact wave from state B # (a) to state B #0 (a R ) ∈ G using ϕ2 (B # , a R ) The set of such states B #0 forms the curve composite W3→1→3 (U L ) If a = a R , then B = U L0 and B # = U L0# = B #0 If a = a L , then B = U L , B # = U L# and B #0 = U L#0 So, U L#0 and U L0# are two endpoints of this curve The curve of composite waves (U L ) is defined as follows (U L ) := W3→1 (U L ) ∪ W1→3 (U L ) ∪ W3→1→3 (U L ) (3.1) Thus, (U L ) has two endpoints I and U−0 = (h 0− , u 0− ) ∈ G − (Fig 1) As mentioned above, the first part of (U L ), which is W3→1 (U L ) is an arc of the curve W1 (U L0 ), and so it is strictly decreasing as written in the form u = u(h) The third part of (U L ), which is W3→1→3 (U L ) is an arc of the hyperbola u/ h = positive constant, so it is also strictly decreasing as written in the form u = u(h) as well Now, we consider the monotone property of the composite curve W1→3 (U L ) As seen in Sect 2, given a state U0 = (h , u ) at topography level a0 , the state U = (h, u) at topography level a = a0 that can be connected to U0 by a stationary wave satisfies the equations F(h; U0 ) = 2gh + (2g(a − a0 − h ) − u 20 )h + h 20 u 20 = 0, 123 u= u0h0 (3.2) h Properties of the Wave Curves in the Shallow Water 323 B (U ) Fig The composite curve W2←3 R Similar to Lemma 3.1, we have following lemma Lemma 3.5 Let U R ∈ G ∪ C+ ∪ G + , a R > a L , and let U2 (h) = (h, u = w2B (U R ; h)) ∈ W2B (U L ) ∩ G , where w2B is defined by (2.1) and (h ± , u ± ) = W1 (U L ) ∩ C± Then, the function ϕ2 (h) = ϕ2 (U2 (h), a L ), h ∈ [h − , h + ] is strictly increasing Proof The function ϕ2 (h) satisfies the equation G(h) := F(h; U1 (h)) = 2gϕ23 (h) 2 + 2g(a L − a R − h) − w2B (h) ϕ2 (h)2 + h w2B (h) = (3.15) = ϕ2 (h)A + B, (3.16) Therefore, where A = ϕ22 (h) 3gϕ2 (h) + 2g(a L − a R − h) − w2B (h) , B = ϕ2 (h) hw2B (h) − gϕ22 (h) + w2B (h)w2B (h) h − ϕ22 (h) To establish the monotony of ϕ2 , we will show that A > 0, B < 123 324 M D Thanh, D H Cuong Indeed, it holds that A = 3gϕ23 (h) − 2gϕ23 (h) + h w2B (h) = gϕ23 (h) − h w2B (h) 2 > gh − h w2B (h) = h gh − w2B (h) ≥ 0, since U2 (h) = (h, w2B (h)) ∈ G Next, to prove that B < 0, we need to show that θ (h) := w2B (h)w2B (h) ≥ −g, h − ≤ h ≤ h + (3.17) Indeed, if w2B (h) > 0, then θ (h) = w2B (h)w2B (h) > > −g So, we remain to prove (3.17) for the case w2B (h) ≤ This means that h − ≤ h ≤ h , where h is the h-intercept of the backward curve W2B (U R ) It holds that d w2B (h)w2B (h) = w2B + w2B (h)w2B (h) > 0, dh for h − ≤ h ≤ h , since the curve W2B (U R ) is strictly concave This implies that the function θ (h) is increasing So, θ (h) ≥ θ (h − ) = − gh − w2B (h − ) = − gh − g = −g, h − ≤ h ≤ h (3.18) h− which establishes (3.17) Thus, B = ϕ2 (h) hw2B (h) − gϕ22 (h) + w2B (h)w2B (h) h − ϕ22 (h) < ϕ2 (h) hw2B (h) − gϕ22 (h) + g ϕ22 (h) − h 2 = ϕ2 (h)h w2B (h) − gh < 0, by (3.17) and U2 (h) = (h, w2B (h)) ∈ G Thus, A > 0, B < and therefore ϕ2 (h) = d ϕ2 (U2 (h), a L ) > dh This terminates the proof of Lemma 3.5 Lemma 3.6 If U R ∈ G ∪ C+ ∪ G + and a R > a L , the function u(h) = w2B (h)h/ϕ2 (U2 (h), a L ) is strictly increasing with respect to h ∈ [h − , h + ], where U2 (h) = (h, u = w2B (U R ; h)) ∈ W2B (U R ) ∩ G and w2B is defined by (2.1) 123 Properties of the Wave Curves in the Shallow Water 325 Proof First, consider the case w2B (h) ≤ We will show that the function f (h) = w2B (h)h is strictly increasing with respect to h ∈ [h − , h ] Actually, since w2B (h) = √ √ √ u R + g h − h R , we have f (h) = w2B (h)h + w2B (h) = g h + w2B (h) = h gh + w2B (h) (3.19) It holds that f (h) = g + h g = h g , h which is positive for h > So, the function f is strictly increasing on the interval h ∈ (h − , h ), where h is the h-intercept of the backward curve W2B (U R ) Moreover, f (h − ) = gh − + w2B (h − ) = gh − − gh − = From (3.19), the last inequality shows that f (h) > 0, h − < h < h and so f is strictly increasing for h − ≤ h ≤ h Let us now consider the function u(h) = w2B (h)h f (h) = , h− ≤ h ≤ h ϕ2 (U1 (h), a L ) ϕ2 (h) (3.20) It holds that u (h) = f (h)ϕ2 (h) − f (h)ϕ2 (h) ϕ22 (h) > 0, since ϕ2 (h) > 0, f (h) > and f (h) < Next, we consider the case w2B (h) > Observe that u(h) satisfying the equation a R − aL + w2B (h) − u (h) + h − ϕ2 (h) = 2g Differentiating that equation with respect to h, we get w2B (h)w2B (h) − u(h)u (h) + − ϕ2 (h) = 0, g or u(h)u (h) = w2B (h)w2B (h) + g − gϕ2 (h) 123 326 M D Thanh, D H Cuong Multiplying both sides of the last equation by A and using (3.17), we get u(h)u (h)A = w2B (h)w2B (h) + g A − g Aϕ2 (h) = w2B (h)w2B (h) + g A + g B = w2B (h)w2B (h) + g gϕ23 (h) − h w2B (h) + gϕ2 (h) hw2B (h) − gϕ22 (h) + w2B (h)w2B (h) h − ϕ22 (h) 2 = w2B (h)w2B (h)h gϕ2 (h) − w2B (h) + ghw2B (h)(ϕ2 (h) − h) > 0, since w2B (h) > 0, ϕ2 (h) > h and U2 (h) = (h, w2B (h)) ∈ G Since A > and u(h) = w2B (h)h/ϕ2 (h) > by (3.20), the last inequality implies that u (h) > This terminates the proof of Lemma 3.6 From Lemmas 3.5 and 3.6, we obtain the monotonicity of the composite wave curve B (U ) as follows W2←3 R Theorem 3.7 If U R ∈ G ∪ C+ ∪ G + and a R > a L , then the composite curve B (U ) can be parameterized by h-component in the form u = u(h), ˜ h− ≤ h ≤ W2←3 R ˜ is strictly increasing function of h − ≤ h ≤ h + h + , where u = u(h) Deterministic Existence for the Riemann Problem 4.1 Case : U L ∈ G ∪ C+ and a L > a R Let us denoted by J = 0, w2B (U R , 0) the intersection point of the curve W2B (U R ) : u = w2B (U R , h) and the u-axis If w2B (U R , 0) < u up , then I = (0, u up ) is located above the curve W2B (U R ) If u 0− < w2B (U R , h 0− ) then U−0 = (h 0− , u 0− ) is located below the curve W2B (U R ) Whenever these two conditions are met, the backward wave curve W2B (U R ) intersects the composite curve (U L ) This leads to the existence of solutions of the Riemann problem, as stated in the following theorem Theorem 4.1 Let w2B and w3 be given by (2.1) and (2.13), respectively Assume that the left-hand state U L ∈ G ∪ C+ , a L > a R , and the right-hand state U R satisfies w2B (U R , 0) < u up , w2B (U R , h 0− ) > u 0− , (4.1) where u up = u 0L + gh 0L , U L0 = (h 0L , u 0L ) = (ϕ1 (U L , a R ), w3 (U L , ϕ1 (U L , a R ))) , U− (h − , u − ) = W1 (U L ) ∩ C− , U−0 = (h 0− , u 0− ) = (ϕ2 (U− , a R ), w3 (U− , ϕ2 (U− , a R ))) 123 (4.2) Properties of the Wave Curves in the Shallow Water 327 Then, the Riemann problems (1.1)–(1.2) has a solution Proof Let the curve (U L ) be parameterized as h = h(m), u = u(m), for m in some interval I Since I, U−0 are the two endpoints of (U L ), we can assume without loss of generality that I = [m , m ], where the values m , m are such that I = (h(m ), u(m )) , and U−0 = (h(m ), u(m )) We define a function G(m) := u(m) − w2B (U R , h(m)), m ∈ I it is easy to see that the function G(m) is continuous on I = [m , m ] Moreover, it holds that G(m ) = u(m ) − w2B (U R , h(m )) = u up − w2B (U R , 0) > 0, and G(m ) = u(m ) − w2B (U R , h(m )) = u 0− − w2B (U R , h 0− ) < Therefore, there exists a value m ∈ (m , m ) such that G(m ) = 0, by IntermediateValue Theorem This means that the curve (U L ) intersects the curve W2B (U R ) at a point corresponding to m , see Fig First, if the curve (U L ) intersects the part W3→1 (U L ), then the Riemann problem has a solution of the form W3 U L , U L0 ⊕ W1 U L0 , U ⊕ W2 (U, U R ) Second, if the curve (U L ) intersects the part W3→1→3 (U L ), then the Riemann problem has a solution of the form W3 U L , B ⊕ W1 B, B # ⊕ W3 B # , B #0 ⊕ W2 B #0 , U R Finally, if the curve (U L ) intersects the part W1→3 (U L ), then the Riemann problem has a solution of the form W1 U L , A ⊕ W3 A, A0 ⊕ W2 A0 , U R This completes the proof of Theorem 4.1 123 328 M D Thanh, D H Cuong Fig The composite curve (U L ) intersects the backward curve W2B (U R ) Consider the forward wave curve W2 (U0 ) A point is located above or below the curve W2 (U0 ) can be characterized as in the following lemma Lemma 4.2 Given two states Ui = (h i , u i ), i = 1, Then, u < w2B (U2 , h ) if and only if u > w2 (U1 , h ) This means that a state U1 is located below the curve W2B (U2 ) if and only if U2 is located above the curve W2 (U1 ) in the (h, u)-plane Proof The state U1 is located below the curve W2B (U2 ) means that u < w2B (U2 , h ) , or u1 < √ √ √ u2 + g h1 − h2 , h1 ≤ h2, u2 + g (h − h ) h1 + h2 , h1 > h2 The last inequalities can be equivalently re-written as u2 > 123 √ √ √ u1 + g h2 − h1 , h2 ≥ h1, u1 + g (h − h ) h2 + h1 , h2 < h1, Properties of the Wave Curves in the Shallow Water 329 which means that u > w2 (U1 , h ) This completes the proof of Lemma 4.2 As seen by Theorem 4.1, a large neighborhood of the left-hand state U L in which the right-hand state U R can be chosen is obtained for the existence It is interesting that this region contains also large regions with simpler geometry such as triangles and rectangles This enables us to see more clearly the existence domain Let us now describe these regions Since the curve W2 (U−0 ) is strictly concave, then the tangent d of the curve W2 (U−0 ) at U−0 is always above the curve W2 (U−0 ) The tangent d is given by g u = u bottom (h) := u 0− + h − h 0− (4.3) h 0− The tangent d intersects the line u = u up , where u up is given by (4.2), in the (h, u)plane at a point with h = h end := h 0− + h 0− u up − u 0− g (4.4) We define a triangle T as follows: T = (h, u)| < h < h end , u bottom (h) < u < u up , (4.5) see Fig Corollary 4.3 Assume that U L ∈ G ∪ C+ , a L > a R Then, U L ∈ T defined by (4.5), and whenever U R ∈ T , the Riemann problem (1.1)–(1.2) has a solution Proof Since U R ∈ T , then U R is located above the tangent d, therefore, above the forward curve W2 (U−0 ) According to lemma 4.2, U−0 is located below the backward curve √ W2B (U R ), therefore, u 0− < w2B (U R , h 0− ) Furthermore, w2B (0) = u R − gh R < u R < u up According to theorem 4.1, the Riemann problem has a solution Now, we proof U L ∈ T Since w1 (U L , h) is strictly decreasing function and U L ∈ G , U− ∈ G , then < h L < h − < ϕ2 (U− , a R ) = h 0− < h end Other hand, since h 0L = ϕ1 (U L , a R ) < h L , then u bottom (h L ) = u 0− + g uLhL (h L − h 0− ) < < u L < h− h 0L = u 0L < u 0L + gh 0L = u up 123 330 M D Thanh, D H Cuong Fig The triangle T Thus, U L ∈ T We define the rectangle R as follows: R = {(h, u)| < h < h 0− , u 0− < u < u up }, (4.6) see Fig Then, it holds for every (h, u) ∈ R that < h < h 0− < h end , u bottom (h) = u 0− + g (h − h 0− ) < u 0− < u < u up h 0− This implies that R ⊂ T Moreover, we can check easily that U L ∈ R This establishes the following corollary 123 Properties of the Wave Curves in the Shallow Water 331 Fig The rectangle R Corollary 4.4 Consider the case where the left-hand state U L ∈ G ∪ C+ , a L > a R Then, U L ∈ R defined by (4.6) Whenever U R ∈ R, i.e, < h R < h 0− , u 0− < u R < u up , the Riemann problem (1.1)–(1.2) admits a solution From the proof of Theorem 4.1, the monotone property of the wave curves and the monotonicity of the curves of composite waves as seen in Sect 3, we obtain the following results on the existence and uniqueness of the Riemann solutions Corollary 4.5 Assume that U L ∈ G ∪C+ , a L > a R , and U L# ∈ W1 (U L )∩(G + ∪C+ ) is the state such that λ1 (U L , U L# ) = Let U L#0 be the state obtained from U L# by an admissible stationary jump from the level a L to a R , i.e., #0 # # # U L#0 = (h #0 L , u L ) = ϕ2 (U L , a R ), w3 (U L , ϕ2 (U L , a R )) 123 332 M D Thanh, D H Cuong (a) If the right-hand state U R satisfies < u #0 w2B U R , h #0 L L , w2B (h 0− ) > u 0− , the Riemann problem for (1.1)–(1.2) has a unique solution form W1 (U L , A) ⊕ W3 (A, A0 ) ⊕ W2 (A0 , U R ) (b) If the right-hand state U R satisfies w2B (U R , 0) < u up , > u 0# w2B h 0# L L , where u up is defined by (4.2), then, the Riemann problem for (1.1)–(1.2) has a unique solution form W3 U L , U L0 ⊕ W1 U L0 , U ⊕ W2 U, U R Corollary 4.6 Assume that U L ∈ G ∪ C+ , a L > a R (a) If the right-hand state U R satisfies h #0 L < h R < h−, u 0− < u R < u #0 L , where U− ∈ W1 (U L ) ∩ C− , and U−0 = (h 0− , u 0− ) ∈ G is the state obtained by a stationary jump from U− , then, the Riemann problem for (1.1)–(1.2) has a unique solution form W1 U L , A ⊕ W3 A, A0 ⊕ W2 A0 , U R (b) If the right-hand state U R satisfies < h R < h 0# L , u 0# L < u R < u up , where u up is defined by (4.2), then, the Riemann problem for (1.1)–(1.2) has a unique solution form W3 U L , U L0 ⊕ W1 U L0 , U ⊕ W2 U, U R 123 Properties of the Wave Curves in the Shallow Water 333 Proof (a) Since h #0 L < h R , then √ #0 #0 u h #0 L = uR + g hL − hR < uR < uL , Moreover, since h R < h 0− , then u h 0− = u R + g h − hR − 1 + > u R > u 0− hR h 0− According to corollary 4.5, the Riemann problem for (1.1)–(1.2) has a unique solution form W1 U L , A ⊕ W3 A, A0 ⊕ W2 A0 , U R (b) Similar to (a) 4.2 Case : U L ∈ G + and a L > a R The above argument gives us the following theorem, whose proof is omitted, since the proof is similar to the one of Theorem 4.1 The illustration for the intersection of the curve of composite waves (U L ) with the backward curve W2B (U R ) is given by Fig Theorem 4.7 Let w2B and w3 be given by (2.1) and (2.13), respectively Assume that the left-hand state U L ∈ G + , a L > a R , and the right-hand state U R satisfies w2B (U R , 0) < u up , w2B (U R , h 0− ) > u 0− , (4.7) where U± = (h ± , u ± ) = W1 (U L ) ∩ C± , u up = u 1+ + gh 1+ , U+1 = h 1+ , u 1+ = (ϕ1 (U+ , a R ), w3 (U+ , ϕ1 (U+ , a R ))) , U−0 = h 0− , u 0− = (ϕ2 (U− , a R ), w3 (U− , ϕ2 (U− , a R ))) (4.8) Then, the Riemann problem (1.1)–(1.2) admits a solution From Theorem 4.7 and the monotonicity of the W1→3→1 (U L ), W2B (U R ), we deduce the following results curves W1→3 (U L ), 123 334 M D Thanh, D H Cuong Fig The composite curve (U L ) intersects the backward curve W2B (U R ) Corollary 4.8 Assume that U L ∈ G + , aL > a R (a) If the right-hand state U R satisfies w2B U R , h 0+ < u 0+ , w2B U R , h 0− > u 0− , where U±0 = (h 0± , u 0± ) is given by (4.8), then, the Riemann problem has a unique solution form W1 (U L , U ) ⊕ W3 U, U ⊕ W2 U , U R (b) Let U+1 and u up be given by (4.8), and let the right-hand state U R satisfy w2B (U R , 0) < u up , 1# w2B h 1# + > u+ , where U+1# ∈ W1 (U+1 ) is the state such that λ1 (U+1 , U+1# ) = Then, the Riemann problem has a unique solution of the form W1 U L , U+ ⊕ W3 U+ , U+1 ⊕ W1 U+1 , U ⊕ W2 (U, U R ) 123 Properties of the Wave Curves in the Shallow Water 335 Corollary 4.9 Assume that U L ∈ G + , a L > a R Using the same notations as in Corollary 4.8 (a) If the right-hand state U R satisfies h 0+ < h R < h 0− , u 0− < u R < u 0+ , the Riemann problem has a unique solution form W1 (U L , U ) ⊕ W3 (U, U ) ⊕ W2 (U , U R ) (b) If the right-hand state U R satisfies < h R < h 1# +, u 1# + < u R < u up , then, the Riemann problem has a unique solution form W1 (U L , U+ ) ⊕ W3 U+ , U+1 ⊕ W1 U+1 , U ⊕ W2 U, U R , 4.3 Case : U R ∈ G ∪ C+ ∪ G + and a R > a L As seen before, the curve of 1-waves W1 (U L ) is strictly decreasing Moreover, Lemma 3.6 also provides us with the monotonicity property of the backward curve of composite B (U ) Thus, we can arrive at the following theorem waves W2←3 R Theorem 4.10 Let w1 and w3 be defined as in (2.1) and (2.13), respectively Assume that U R ∈ G ∪ C+ ∪ G + , a R > a L , and the left-hand state U L satisfies w1 U L , h 0+ < u 0+ , w1 U L , h 0− > u 0− , (4.9) where U± = (h ± , u ± ) = W2B (U R ) ∩ C± , U±0 = (h 0± , u 0± ) = (ϕ2 (U± , a L ), w3 (U± , ϕ2 (U± , a L ))) (4.10) Then, the Riemann problem has a unique solution form W1 (U L , U ) ⊕ W3 (U, U ) ⊕ W2 (U , U R ) B (U ) and The illustration for the intersection of the curve of composite waves W2←3 R the curve W1 (U L ) is given by Fig 123 336 M D Thanh, D H Cuong B (U ) intersects the curve W (U ) Fig The composite curve W2←3 L R Corollary 4.11 Assume that U R ∈ G ∪ C+ ∪ G + , a R > a L , and the left-hand state U L satisfies h 0− < h L < h 0+ u 0− < u L < u 0+ , where U±0 = (h 0± , u 0± ) is given by (4.10) Then, the Riemann problem has a unique solution form W1 (U L , U ) ⊕ W3 (U, U ) ⊕ W2 (U , U R ) Conclusions Hyperbolic systems of balance laws in nonconservative form possess very interesting but rather complicated phenomena In particular, characteristic speeds may coincide and the order in Riemann solutions of elementary waves such as shocks, rarefaction waves, and stationary contacts may be changed from one region to another This can be dealt with when solving the Riemann problem by building up the curves of composite waves In this paper we first establish the monotonicity property of these curves of composite waves Hence, together with earlier works, the monotonicity property of all the wave curves involved in the solving of the Riemann problem for the shallow water equations with discontinuous topography are observed Second, we determine explicitly the domains for the existence and the uniqueness of the Riemann problem The future works in this direction may be involved in the monotonicity properties of the composite wave curves in the more complicated models of hyperbolic 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[15] It is interesting that the shock speeds in the nonlinear characteristic fields may coincide with the characteristic speed of the linearly degenerate field as stated in the following lemma