15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 787 For hyperbolic systems (15.14.4.32), equation (15.14.4.35) has two different real roots: λ 1,2 = 1 2 (p 1 + q 2 ) 1 2 (p 1 – q 2 ) 2 + 4p 2 q 1 .(15.14.4.36) For each eigenvalue λ m of (15.14.4.36), from (15.14.4.34) we find the associated eigen- function b (m) 1 = ϕ m (λ m – q 2 ), b (m) 2 = ϕ m q 1 ,(15.14.4.37) where ϕ m = ϕ m (u, w) is an arbitrary function (it will be defined later); m = 1, 2. From (15.14.4.33), in view of (15.14.4.34), we obtain two equations for the two roots (15.14.4.36): b (m) 1 ∂u ∂t + λ m ∂u ∂x + b (m) 2 ∂w ∂t + λ m ∂w ∂x = 0, m = 1, 2.(15.14.4.38) The function ϕ m in (15.14.4.37) can be determined from the conditions b (m) 1 = ∂R m ∂u , b (m) 2 = ∂R m ∂w .(15.14.4.39) On differentiating the first relation (15.14.4.39) with respect to w andthe second with respect to u, we equate the mixed derivatives (R m ) uw and (R m ) wu . In view of (15.14.4.37), we obtain the following linear first-order partial differential equation for ϕ m : ∂ ∂w [ϕ m (λ m – q 2 )] = ∂ ∂u (ϕ m q 1 ). (15.14.4.40) This equation can be solved by the method of characteristics; see Subsection 13.1.1. Assum- ing that a solution of equation (15.14.4.40) has been obtained (any nontrivial solution can be taken) and taking into account formulas (15.14.4.37), we find the functions R m = R m (u, w) from system (15.14.4.39). Replacing b (m) 1 and b (m) 2 in (15.14.4.38) by the right-hand sides of (15.14.4.39), we get two equations: ∂R 1 ∂t + λ 1 (R 1 , R 2 ) ∂R 1 ∂x = 0, ∂R 2 ∂t + λ 2 (R 1 , R 2 ) ∂R 2 ∂x = 0, (15.14.4.41) where λ m (R 1 , R 2 )=λ m (u, w), m = 1, 2. The functions R 1 and R 2 appearing in system (15.14.4.41) are called Riemann invariants. System (15.14.4.41) admits two exact solutions: R 1 = C 1 , x – λ 2 (C 1 , R 2 )t = Φ 1 (R 2 ), R 2 = C 2 , x – λ 1 (R 1 , C 2 )t = Φ 2 (R 1 ), where C m are arbitrary constants and Φ m (R 3–m ) are arbitrary functions (m = 1, 2). If the function λ 1 in (15.14.4.41) is independent of R 2 , then the solution of system (15.14.4.41) is reduced to successive integration of two quasilinear first-order partial dif- ferential equations. 788 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Remark 1. Sometimes it is more convenient to use the formulas b (m) 1 = ϕ m p 2 , b (m) 2 = ϕ m (λ m – p 1 )(15.14.4.37a) rather than (15.14.4.37). In this case, equation (15.14.4.40) is replaced by ∂ ∂w (ϕ m p 2 )= ∂ ∂u [ϕ m (λ m – p 1 )]. (15.14.4.40a) Remark 2. The Riemann functions are not uniquely defined. Transformations of the dependent variables R 1 = Ψ 1 (z 1 ), R 2 = Ψ 2 (z 2 ), where Ψ 1 (z 1 )andΨ 2 (z 2 ) are arbitrary functions, preserve the general form of system (15.14.4.41); what changes are the functions λ m only. This is due to the functional arbitrariness in determining the functions ϕ m from equations (15.14.4.40) or (15.14.4.40a). This arbitrariness can be used for the selection of most simple Riemann functions. Example 4. Consider the system of equation of longitudinal oscillations of an elastic bar (15.14.4.10), which is a special case of system (15.14.4.32) with p 1 = 0, q 1 =–1, p 2 =–σ (u), q 2 = 0.(15.14.4.42) To determine the eigenvalues and the corresponding eigenfunctions, we use formulas (15.14.4.36), (15.14.4.37), (15.14.4.42) and equation (15.14.4.40). As a result, we obtain λ 1,2 = σ (u); b (m) 1 = σ (u), b (m) 2 = 1, ϕ m = 1 (m = 1, 2). The Riemann functions are found from equations (15.14.4.39): R 1 =–w + σ (u) du, R 2 = w + σ (u) du. Note that the Riemann functions coincide, up to notation, with the integrals (15.14.4.21) obtained earlier from other considerations. Example 5. Consider the system of equations (15.14.4.26) describing one-dimensional flow of an ideal adiabatic gas. First, reduce the system to the canonical form ∂ρ ∂t + v ∂ρ ∂x + ρ ∂v ∂x = 0, ∂v ∂t + p (ρ) ρ ∂ρ ∂x + v ∂v ∂x = 0. This is a special case of system (15.14.4.32) with u = ρ, w = v, p 1 = v, q 1 = ρ, p 2 = p (ρ)/ρ, q 2 = v.(15.14.4.43) To determine the eigenvalues and the corresponding eigenfunctions, we use formulas (15.14.4.36), (15.14.4.37), (15.14.4.43) and equation (15.14.4.40). As a result, we obtain λ 1,2 = v p (ρ); b (m) 1 = √ p (ρ) ρ , b (m) 2 = 1, ϕ m = 1 ρ (m = 1, 2). The Riemann functions are found from equations (15.14.4.39): R 1 = v + √ p (ρ) ρ dρ, R 2 =–v + √ p (ρ) ρ dρ. 3 ◦ . Let us show that the Riemann invariants are constant along the rarefaction waves for hyperbolic systems. The substitution of the self-similar solution forms R m = R m (ξ), where ξ = x/t, into system (15.14.4.41) results in the following system of two ordinary differential equations: ξ – λ m (R 1 , R 2 ) dR m dξ = 0 (m = 1, 2). (15.14.4.44) The equality ξ = λ 1 (R 1 , R 2 ) takes place along the first rarefaction wave. Hence, the first factor in the second equation of (15.14.4.44) is nonzero. Therefore, the second factor in the second equation of (15.14.4.44) is zero. It follows that R 2 = const along the first rarefaction wave. Along the second rarefaction wave, R 1 is constant. 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 789 15.14.4-7. Hyperbolic n × n systems of conservation laws. Exact solutions. Here we consider systems of conservation laws of the form ∂F(u) ∂t + ∂G(u) ∂x = 0,(15.14.4.45) where u = u(x, t) is a vector function of two scalar variables, and F = F(u)andG = G(u) are vector functions, u =(u 1 , , u n ) T , u i = u i (x, t); F =(F 1 , , F n ) T , F i = F i (u); G =(G 1 , , G n ) T , G i = G i (u). Here and henceforth, (u 1 , , u n ) T stands for a column vector with components u 1 , , u n . Note three important types of exact solutions to system (15.14.4.45): 1. For any F and G, system (15.14.4.45) admits the following simplest solutions: u = C,(15.14.4.46) where C is an arbitrary constant vector. 2. System (15.14.4.45) admits self-similar solutions of the form u = u(ξ), ξ = x/t.(15.14.4.47) The procedure for finding them is analogous to that used in Paragraph 15.14.4-2 (see also Paragraph 15.14.4-10). 3. System (15.14.4.45) also admits more general exact solutions of the form u 1 = u 1 (u n ), , u n–1 = u n–1 (u n ), (15.14.4.48) where all the components are functionally related and can be expressed in terms of one of them. After substituting (15.14.4.48) into the original system (15.14.4.45), we require that all the resulting equations coincide. As a result, we obtain a system of (n – 1)ordinary differential equations for determining the dependences (15.14.4.48) and one first-order partial differential equation for u n = u n (x, t). The mentioned procedure is described in detail in Paragraph 15.14.4-3 for the case of a two-equation system. Let us show that some systems of conservation laws can be represented as systems of ordinary differential equations along curves x = x(t) called characteristic curves. Differentiating both sides of system (15.14.4.45) yields ∂u ∂t + A ∂u ∂x = 0,(15.14.4.49) where A = F –1 (u) G(u), F(u) is the matrix with entries ∂F i ∂u j , G(u) is the matrix with entries ∂G i ∂u j ,and F –1 is the inverse of the matrix F. Let us multiply each scalar equation in (15.14.4.49) by b i = b i (u) and take the sum. On rearranging terms under the summation sign, we obtain n i=1 b i ∂u i ∂t + n i,j=1 b j a ji ∂u i ∂x = 0,(15.14.4.50) where a ij = a ij (u) are the entries of the matrix A. 790 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS If b =(b 1 , , b n ) is a left eigenvector of the matrix A(u) that corresponds to an eigenvalue λ = λ(u), so that n j=1 b j a ji = λb i , then equation (15.14.4.50) can be rewritten in the form n i=1 b i ∂u i ∂t + λ ∂u i ∂x = 0.(15.14.4.51) Thus, system (15.14.4.49) is transformed to a linear combination of total derivatives of the unknowns u i with respect to t along the direction (λ, 1) on the plane (x, t), i.e., the total time derivatives are taken along the trajectories having the velocity λ: n i=1 b i du i dt = 0, dx dt = λ,(15.14.4.52) where b i = b i (u), λ = λ(u), x = x(t), du i dt = ∂u i ∂t + dx dt ∂u i ∂x . Equations (15.14.4.52) are called differential relations on characteristics. The second equation in (15.14.4.52) explains why an eigenvalue λ is called a characteristic velocity. System (15.14.4.49) is called hyperbolic if the matrix A has n real eigenvalues and n linearly independent left eigenvectors. If all eigenvalues are distinct for any u, then system (15.14.4.49) is said to be strictly hyperbolic. Remark 1. If the hyperbolic system (15.14.4.49) is linear and the coefficients of the matrix A are constant, then the eigenvalues λ k are constant and the characteristic lines in the (x, t) plane become straight lines: x = λ k t + const. Since all eigenvalues λ k are different, the general solution of system (15.14.4.49) can be represented as the sum of particular solutions as follows: u = φ 1 (x – λ 1 t)r 1 + ···+ φ n (x – λ n t)r n , where the φ k (ξ k ) are arbitrary functions, ξ k = x –λ k t,andr k is the right eigenvector of A corresponding to the eigenvalue λ k , k = 1, , n. The particular solutions u k = φ k (x – λ k t)r k are called traveling-wave solutions. Each of these solutions represents a wave that travels in the r k -direction with velocity λ k . Remark 2. The characteristic form (15.14.4.51) of the hyperbolic system (15.14.4.49) forms the basis for the numerical characteristics method which allows the solution of system (15.14.4.49) in its domain of continuity. Suppose that we already have a solution u(x, t) for all values of x and a fixed time t. To construct a solution at a point (x, t + Δt), we find the points (x – λ k Δt, t) from which the characteristics arrive at the point (x, t + Δt). Since the u(x – λ k Δt, t) are known, relations (15.14.4.52) can be regarded as a system of n linear equations in the n unknowns u(x, t + Δt). Thus, a solution for the time t + Δt can be found. 15.14.4-8. Shock waves. Rankine–Hugoniot jump conditions. Let us consider a discontinuity along a trajectory x f (t) and obtain the mass balance condition along a discontinuity (shock wave). The region x > x f (t) is conventionally assumed to lie ahead of the shock, and the region x < x f (t) is assumed to lie behind the shock. The shock speed D is determined by the relation D = dx f dt .(15.14.4.53) 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 791 To refer to the value of a quantity, A, behind the shock, the minus superscript will be used, A – , since this value corresponds to negative x in the initial value formulation. Likewise, the value of A ahead of the shock will be denoted A + . For an arbitrary quasilinear system in the form of conservation laws (15.14.4.45), the balance equations for a shock can be represented as [F i (u)]D =[G i (u)], i = 1, , n,(15.14.4.54) where [A]=A + –A – stands for the jump of a quantity A at the shock. Formulas (15.14.4.54) are called the Rankine–Hugoniot jump conditions; they are derived from integral conse- quences of the equations in question in much the same ways as for a single equation (see Paragraph 13.1.3-4). Example 6. The Rankine–Hugoniot conditions for an isentropic gas flow (15.14.4.26) follow from (15.14.4.54). We have [ρ]D =[ρv], [ρv]D =[ρv 2 + p(ρ)]. (15.14.4.55) Here, the density and the velocity ahead of the shock are denoted ρ + and v + , while those behind the shock are ρ – and v – . Eliminating the shock speed D from (15.14.4.55), we obtain the relation v + – v – = (ρ + – ρ – ) p(ρ + )–p(ρ – ) ρ – ρ + .(15.14.4.56) Each of the signs before the square root in (15.14.4.56) corresponds to a branch of the locus of points that can be connected with a given point (v – , ρ – ) by a shock (see Fig. 15.9a). For an ideal polytropic gas, one should set p = Aρ γ in formulas (15.14.4.55) and (15.14.4.56). Let us determine the set of states (v + , ρ + ) reachable by a shock from a given point (v – , ρ – ). Express the point (v + , ρ + ) ahead of the shock via the solution of the transcendental system (15.14.4.55) to obtain v + = v + (v – , ρ – , D), ρ + = ρ + (v – , ρ – , D). The graphs of the solution determined by (15.14.4.56) are shown in Figs. 15.9a and 15.9b. The solid lines correspond to the minus sign before the radical and represent stable (evolutionary) shocks, while the dashed lines correspond to the plus sign and represent unstable (nonevolutionary) shocks; see below. ()a ()b v - v + r - r + v - v + r - r + Figure 15.9. Loci of points that can be connected by a shock wave: (a) with a given state (v – , ρ – )and(b) with a given state (v + , ρ + ). The solid lines correspond, respectively, to the minus and plus signs in formula (15.14.4.56) (for p = Aρ γ ) before the radical for cases (a)and(b). 792 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 15.14.4-9. Shock waves. Evolutionary conditions. Lax condition. In general, discontinuities of solutions are surfaces where conditions are imposed that relate the quantities on both sides of the surfaces. For hyperbolic systems in the conservation-law form (15.14.4.45), these relations have the form (15.14.4.54) and involve the discontinuity velocity D. The evolutionary conditions are necessary conditions for unique solvability of the prob- lem of the discontinuity interaction with small perturbations depending on the x-coordinate normal to the discontinuity surface. For hyperbolic systems, a one-dimensional small per- turbation can be represented as a superposition of n waves, each being a traveling wave propagating at a characteristic velocity λ i . This allows us to classify all these waves into in- coming and outgoing ones, depending on the sign of the difference λ i –D. Incoming waves are fully determined by the initial conditions, while outgoing ones must be determined from the linearized boundary conditions at the shock. We consider below the stability of a shock with respect to a small perturbation. This kind of stability is determined by incoming waves. For this reason, we focus below on incoming waves. Let m + and m – be the numbers of incoming waves from the right and left of the shock, respectively. It can be shown that if the relation m + + m – – 1 = n (15.14.4.57) holds, the problem of the discontinuity interaction with small perturbations is uniquely solvable. Relation (15.14.4.57) is called the Lax condition. If (15.14.4.57) holds, the corresponding discontinuity is called evolutionary; otherwise, it is called nonevolutionary. For evolutionary discontinuities, small incoming perturbations generate small outgoing perturbations and small changes in the discontinuity velocity. If m + + m – – 1 > n, then either such discontinuities do not exist or the perturbed quantities cannot be uniquely determined (the given conditions are underdetermined). If m + + m – – 1 < n, then the problem of the discontinuity interaction with small perturbations has no solution in the linear approximation. Previous studies of various physical problems have shown that the interaction of nonevolutionary discontinuities with small perturbations results in their disintegration into two or more evolutionary discontinuities. The evolutionary condition (15.14.4.57) can be rewritten in the form of inequalities relating the shock speed D and the velocities λ i of small disturbances. Let us enumerate the characteristic velocities on both sides of the discontinuity so that λ 1 (u) ≤ λ 2 (u) ≤ ···≤ λ n (u). Ashockiscalledak-shock if both kth characteristics are incoming: if i > k,thenD < λ i ; if i < k,thenD > λ i ; if i = k,thenλ + i < D < λ – i . (15.14.4.58) 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 793 Below is another, equivalent statement of the Lax condition: n + 1 inequalities out of the 2n inequalities λ + k ≤ D ≤ λ – k (k = 1, , n)(15.14.4.59) must hold. For two-equation systems, the shock evolutionarity requires that three out of four in- equalities λ + 1 ≤ D ≤ λ – 1 , λ + 2 ≤ D ≤ λ – 2 ,(15.14.4.60) must hold. Example 7. The Lax condition for the adiabatic gas flow equations (15.14.4.26) are obtained by sub- stituting the eigenvalue expressions λ 1,2 = v √ p (ρ) (see Example 5) into inequalities (15.14.4.60). We have v + p (ρ + )<D < v – p (ρ – ). (15.14.4.61) The shock evolutionarity requires that three of the four inequalities in (15.14.4.61) hold. Substituting the equation of state for a polytropic ideal gas, p = Ar γ , into (15.14.4.61), we obtain the following evolutionarity criterion: v + Aγ(ρ + ) γ–1 < D < v – Aγ(ρ – ) γ–1 .(15.14.4.62) For the adiabatic gas flow system (15.14.4.26), the solution vector is u =(ρ, v) T . Figure 15.9a shows the locus of points u + that can be connected by a shock to the point u – ; it is divided into the evolutionary part (solid line) and nonevolutionary part (dashed line). It can be shown that a shock issuing from the point (v – , ρ – ) and passing through any point (v + , ρ + ) of the solid part of the locus of first-family shocks obeys the Lax conditions (15.14.4.62). The shock speed D of the first family decreases from λ 1 (u – ), for points u + tending to point u – ,tov – as ρ + → 0 and v + → –∞. Along the locus of the second-family shocks, the speed decreases from λ 2 (u – ), for points u + tending to u – ,to–∞ as ρ + →∞and v + → –∞. Figure 15.9b depicts the locus of points u – that can be connected by a shock to the point u + . The evolutionary part of the locus is shown by a solid line; the dashed line shows the nonevolutionary part. The shock speed D of the first family increases from λ 1 (u + ), for points u – tending to u + ,to∞ as ρ – →∞and v – →∞. Along the locus of the second family shocks, the speed increases from λ 2 (u + ), for points u – tending to u + ,tov + as ρ – → 0 and v – →∞. 15.14.4-10. Solutions for the Riemann problem. Solutions describing shock waves. 1 ◦ . Self-similar solutions (15.14.4.47) together with the simplest solutions (15.14.4.46) enable the solution of the Riemann problem. This problem is formulated as follows: find a function u = u(x, t) that solves system (15.14.4.45) for t > 0 and satisfies the following initial condition of a special form: u = u L if x < 0 u R if x > 0 at t = 0.(15.14.4.63) Here, u L and u R are two prescribed constant vectors. With the self-similar variable ξ = x/t, the initial condition (15.14.4.63) is transformed into the boundary conditions u → u L as ξ → –∞, u → u R as ξ →∞.(15.14.4.64) 2 ◦ . Without loss of generality, we will be considering the case F(u)=u. The substitution of the self-similar form u(x, t)=u(ξ) into (15.14.4.45) with F(u)=u yields G – ξI u ξ = 0,(15.14.4.65) where G = G(u) is the matrix with entries G ij = ∂G i ∂u j and I is the identity matrix. . value formulation. Likewise, the value of A ahead of the shock will be denoted A + . For an arbitrary quasilinear system in the form of conservation laws (15.14.4.45), the balance equations for. evolutionary part (solid line) and nonevolutionary part (dashed line). It can be shown that a shock issuing from the point (v – , ρ – ) and passing through any point (v + , ρ + ) of the solid part of the. G i (u). Here and henceforth, (u 1 , , u n ) T stands for a column vector with components u 1 , , u n . Note three important types of exact solutions to system (15.14.4.45): 1. For any F and G, system