2 Introduction T(,t) = = L Figure..: Geometry of the prismatical rod of Eample... T = u(,t) = L T Figure..2: Geometry of the taut string of Eample..2.

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1 Chapter Introduction. Problems Leading to Partial Dierential Equations Many problems in science and engineering are modeled and analyzed using systems of partial dierential equations. Realistic models involve multi-dimensional behavior, nonlinearity, and irregular boundaries which are too complicated to be solved by analytic techniques. Numerical techniques provide useful approimations and now provide the dominant approach to addressing partial dierential equations. Let's begin by eamining eamples of physical situations that involve partial dierential equations. Eample... Consider the problem of determining the temperature T in a long prismatical rod of length L for times t > given that its temperature is known at the instant that the eperiment is started (t = ) and that some information, e.g., the temperature or the heat u, is prescribed at the boundaries = L (Figure..). The temperature in the rod may be determined as a function of the spatial coordinate and the time t by solving the one-dimensional heat conduction equation subject to the initial @T (k ) <<L T ( ) = T () L (..b) and appropriate boundary data, such as T ( t)=t =: (..c)

2 2 Introduction T(,t) = = L Figure..: Geometry of the prismatical rod of Eample... T = u(,t) = L T Figure..2: Geometry of the taut string of Eample..2. The parameters, c, and k are, respectively, the density, specic heat, and conductivity of the rod. Equation (..c) implies that the left end of the rod is maintained at the temperature T L while the right endis insulated. Eample..2. The problem of nding the deection u( t) of a taut string of length L, as shown in Figure..2, given its initial deection and velocity is governed 2 = u <<L t> (..2a) u( ) = u = v () L (..2b) u( t)= u(l t) = t>: (..2c) Here, T is the tension in the string, is the string's density per unit length, u () isthe initial deection, and v () is the initial velocity. Eample..3. The problem of determining the stresses in a prismatical shaft that is subjected to a torque T (Figure..3)is governed by = ;2 ( y) 2 (..3a) 2 = ( y) (..3b)

3 .. Problems Leading to PDEs 3 y Θ T z Figure..3: Geometry of the prismatical rod of Eample T =2G Z Z d! yz : (..3c) The variable ( y) is a stress function. Once it has been determined from (..3a,b), the angle of twist per unit length and the components of the shear stress z and yz may be determined from (..3c). The parameter G is the shear modulus of the rod, is the cross-sectional area of the rod, denotes its boundary. Observe that this problem is in equilibrium and, unlike those of Eamples.. and..2, it does not evolve in time. Eample..4. The two-dimensional ow of a compressible inviscid uid is governed by the Euler equations t + m + n y = (..4a) m t +( m2 + p) +( mn ) y = (..4b) n t +( mn ) +( n2 + p) y = e t +[(e + p) m ] +[(e + p) n ] y =: (..4c) (..4d)

4 4 Introduction y, n, v U, m, u Figure..4: Typical ow region for Eample..4. Subscripts denote partial dierentiation and, m and n, e, and p are the uid density, the and y components of the momentum per unit volume, the total energy per unit volume, and the pressure, respectively. The and y components of the velocity vector, u and v, arerelated to the components of the momentum vector by m = u n = v: (..5) Finally, the pressure must be related to, m, n, and e by an equation of state, which, for a perfect gas, is p =( ; )[e ; (m 2 + n 2 )=2] (..6) where is a constant. Equations (..4a), (..4b), (..4c), and (..4d) epress the physical conditions that the mass, the component of momentum, the y component of momentum, and the energy of the uid, respectively, are conserved during its motion. A typical problem involving the ow around an airfoil is shown in Figure..4. The boundary conditions for this eample would state that the momentum vector must be tangent to the airfoil and that ow conditions far from the airfoil are uniform and prescribed..2 Second-Order Partial Dierential Equations Mathematicians classify partial dierential systems according to the order of their highest derivatives, the number of independent variables, and the number of dependent variables. Other properties, such as linearity, are also used for characterization. Thus, Eamples..,..2, and..3 are second-order, linear, scalar problems in two variables while Eample..4 is a rst-order, nonlinear, vector system in the three variables, y, and

5 .2. Second-Order PDEs 5 t. We'll consider rst-order problems in Section.3 and eamine second-order problems here. In particular, we'll focus on a linear, scalar, partial dierential equation in two variables having the form L[u] au +2bu y + cu yy + du + eu y + fu = g (.2.) where a, b, :::, g are continuous functions of and y and subscripts denote partial dierentiation, e.g., Such second-order equations are classied into three types depending on the roots 2 = ;b p b 2 ; ac a (.2.2a) of the characteristic equation a 2 +2b + c =: (.2.2b) In particular, (.2.) is hyperbolic if, 2 are real and distinct, i.e., ifb 2 ; ac >, elliptic if, 2 are comple, i.e., ifb 2 ; ac <, and parabolic if = 2, i.e., if b 2 ; ac =. Eample.2.. A problem can be of a dierent type in dierent spatial regions. The equation u + u yy = (.2.3) has a =, b =,and c = thus, b 2 ; ac = ; and (.2.3) is hyperbolic in the left-half plane ( <), elliptic in the right-half plane ( >), and parabolic on the line =. There are canonical partial dierential equations of each type that are important for theoretical reasons and when developing practical numerical methods. The canonical hyperbolic equation is (.2.) with a =,c = ;, b = d = e = f = g =,or u ; u yy =: (.2.4)

6 6 Introduction This equation is called the one-dimensional wave equation. Usually one of the independent variables, say y, corresponds to time. Initial and/or boundary conditions have tobe specied for the solution to be uniquely determined. An initial value or Cauchy problem for the wave equation consists of nding u satisfying (.2.4) on y >, ; < <, given the initial conditions u( ) = () u y ( ) = (): (.2.5a) An initial-boundary value problem for the wave equation consists of determining u satisfying (.2.4) on y >, < <, given the initial conditions (.2.5a) and the boundary conditions u( t)+ u ( t)= u( t)+ u ( t)= : (.2.5b) A boundary condition (.2.5b) is called Dirichlet if j = for j = or it is called Neumann if j = and it is called Robin if j and j are nonzero. The canonical parabolic problem is the one-dimensional heat conduction equation which has a =,e = ;, b = c = d = f = g =in (.2.), or u ; u y =: (.2.6) Again, y frequently corresponds to time. The initial value problem for the heat equation is to nd u satisfying (.2.6) on y>, ; <<, given the initial condition u( ) = (): (.2.7a) An initial-boundary value problem for the heat equation is to nd u satisfying (.2.6) on y>, <<, given the initial condition (.2.7a) and the boundary conditions u( t)+ u ( t)= u( t)+ u ( t)= : (.2.7b) The canonical elliptic problem is the two-dimensional Poisson equation where a =, c =,b = d = e = f =in (.2.) to produce u + u yy = g( y): (.2.8)

7 .2. Second-Order PDEs 7 If the function g( y) then (.2.8) is called Laplace's equation. There are no well posed (stable) initial or initial-boundary value problems for Poisson's or Laplace's equation. There are only boundary value problems, which consist of determining u( y) for ( y) 2 satisfying (.2.8) given u + u n = ( y) (.2.9) where n is a unit outward normal vector and,, and are functions of and y. In analogy with the boundary conditions for the wave and heat equations, if the problem is called a Dirichlet problem, if the problem is called a Neumann problem, and if neither nor are zero the problem is called a Robin problem. Eact solutions to each of the canonical problems are known in certain circumstances and the classical technique for obtaining them is by using Fourier series. Fourier series will also play a key role in analyzing nite dierence approimations of the canonical problems, so let us illustrate Fourier's method for a simple heat conduction problem. We're not going to delve into this subject, so consult an elementary dierential equations tet such as Boyce and DiPrima [] for additional information. Eample.2.2. Consider the Dirichlet problem for the heat conduction equation u t = u << t> (.2.a) u( ) = () (.2.b) u( t)=u( t)=: (.2.c) (As noted, y in (.2.6) frequently corresponds to time. In order to emphasize this, we have replaced the symbol y with a t.) The homogeneous boundary conditions suggest the use of the Fourier sine series u( t) = X k= a k (t)sink (.2.) which satises (.2.c) for all k>. Substitution of (.2.) into (.2.a) yields X k= [_a k +(k) 2 a k ]sink =

8 8 Introduction where a superimposed dot denotes time dierentiation. The term in brackets must vanish if the above system is to be satised for all values of k and thus, or _a k +(k) 2 a k = a k (t) =a ke ;k2 2t 8 k> 8 k>: Using this result in (.2.) gives u( t) = X k= a ke ;k2 2t sin k: (.2.2a) The constants a k, k >, are determined from the initial conditions (.2.b) thus, evaluating (.2.2a) at t = yields u( ) = () = X k= a k sin k Multiplying the above equation by sin l, integrating on ( ), interchanging limiting processes, and using the orthogonality properties of the sine function yields Z X Z () sin ld = a k sin k sin ld = a l if k = l 2 otherwise : Thus, k= Z a k =2 ()sinkd 8 k>: (.2.2b) Even without a specic initial function () we are able to discern several properties of the solution. For eample, assuming that (.2.2a) converges, we see that (i) u( t) is a smooth function of and t even if () isn'tand(ii) u( t) is a decreasing function of t. Incidentally, (.2.2) converges in the L 2 norm (cf. (.2.4)) under rather general conditions [3]. To be more specic, suppose that 2 =2 () = 2( ; ) =2 : Then, (.2.2b) implies a k =2[ Z =2 2 sin kd + Z =2 2( ; ) sin kd] = 8 (k) 2

9 .2. Second-Order PDEs 9 and (.2.2a) becomes u( t) = X k= 8 (k) 2 e;k2 2t sin k: (.2.3) With this initial data, (.2.3) is formally converging as O(=k 2 ). In general, the Fourier series converges as O(=k +2 ) if () 2 C. Thus, even (some) discontinuous data leads to convergent Fourier series [3]. It will not be possible to obtain Fourier series or other eplicit solutions to practical (variable-coecient or nonlinear) problems. Nevertheless, we can obtain information about the solution without much eort. Let us, for eample, obtain an estimate of the behavior of the solution in the L 2 norm, which, for (.2.), is dened as ku( t)k 2 Z Multiplying (.2.a) by u and integrating on ( ), we nd This may be written as or, using (.2.4) d 2 dt Z Z uu t d = u 2 d = Z u( t) 2 d =2 : (.2.4) Z uu d: (uu ) d ; d 2 dt ku( t)k2 2 = uu j ; Using the boundary conditions (.2.c) d dt ku( t)k2 2 = ;2 Z Z Z u 2 d: u 2 d: u 2 d Since the integral on the right is non-positive, the L 2 norm of the solution is a decreasing function of t. The decrease is due to the presence of the u term in (.2.a), which we call dissipative. Problems. ([], Section.7.) Find the Fourier-series solution of Laplace's equation u + u yy =

10 Introduction on the rectangle <<a, <y<b, satisfying the boundary conditions u( y)= u(a y) = <y<b u( ) = u( b) =g() <<a: Additionally, ndthe solution in the particular case when < a=2 g() = a ; a=2 <<a : 2. ([5], Section 8.) Find where the following partial dierential equations are hyperbolic, parabolic, and = u =: 3. Under suitable assumptions [2] the equations of two-dimensional, compressible, steady, inviscid ow ((..4) with all time derivatives set to zero) can be reduced to the simpler system (a 2 ; u 2 )u ; uv(u y + v )+(a 2 ; v 2 )v y = v ; u y =: Once again, u and v are the Cartesian coordinates of the velocity vector and a is the speed of sound (a=a ) 2 =; ; 2 u 2 + v 2 with a being the speed of sound when u = v = and > beinga constant. Introducing a potential function ( y) such that a 2 u = v = y

11 .3. Hyperbolic Conservation Laws we satisfy the second partial dierential equation and \reduce" the rst to the transonic full-potential equation (a 2 ; 2 ) ; 2 y y +(a 2 ; 2 y) yy =: Although this is a complicated nonlinear second-order dierential equation, it can still be classied as hyperbolic, parabolic, or elliptic. However, with nonlinear equations, the type may depend on the unknown solution. Determine the type of the transonic full-potential equation. Epress your answer in terms of the Mach number M 2 = u2 + v 2 a 2 = y a 2 : 4. Show that the L 2 norm of the solution of Burgers' equation u t + uu = u << t> where is a positive constant, is a decreasing function of time when the initial and boundary conditions are u( ) = () u( t)=u( t)= t>:.3 Hyperbolic Conservation Laws: Characteristics, Shock Waves, and Rankine-Hugoniot Conditions A conservation laws states that the total amount of some quantity remains unchanged during the evolution of the solution according to the partial dierential equation. In physical processes without dissipation, these quantities might be the total mass, momentum, and energy. In this introductory chapter, let us conne our attention to conservation laws in one space dimension which typically have the form d dt Z ud = ;f(u)j = ;f(u( t)) + f(u( t)) (.3.)

12 2 Introduction where u( t) = u ( t) u 2 ( t). u m ( t) f(u) = f (u) f 2 (u). f m (u) : (.3.2) are density and u vectors, respectively. Equation (.3.) epresses the fact that the rate of change of u within the \volume" is equal to the change in its u through the boundaries =,. If f and u are smooth functions, then (.3.) can be written as Z [u t + f(u) ]d =: Since this result should hold for all possible \control volumes" ( ), the integrand must vanish, i.e., u t + f(u) =: (.3.3) Eample.3.. The Euler equations for one-dimensional compressible inviscid ows may be obtained by setting all y derivatives and n to zero in equations (..4) to obtain t + m = (.3.4a) m t +( m2 + p) = (.3.4b) e t +[(e + p) m ] =: (.3.4c) The pressure is determined by an equation of state which we take in the rather general form p = p( m e). Recall that equations (.3.4a), (.3.4b), and (.3.4c) epress the facts that the mass, momentum, and energy of the uid are neither created nor destroyed and are, hence, conserved. We readily see that the system (.3.4) has the form of (.3.3) with u = 2 4 m e f(u) = 4 m m 2 = + p (e + p)m= 3 5 : (.3.5)

13 .3. Hyperbolic Conservation Laws 3 Eample.3.2. The deection of a taut string has the form u tt = c 2 u (.3.6) where c 2 = T= with T being the tension and being the linear density of the string (cf. (..2)). Equation (.3.6) can be written as a rst-order system in a variety of ways. Perhaps the most common approach is to let u = u t u 2 = cu : (.3.7) Dierentiating with respect to t while using (.3.6) and (.3.7) yields (u ) t = u tt = c 2 u = c(u 2 ) (u 2 ) t = cu t = cu t = c(u ) : Thus, the one-dimensional wave equation has the form of (.3.3) with u ;cu2 u = f(u) = : u 2 ;cu For the remainder of this introductory section, we'll conne our attention to the scalar conservation law u t + f(u) = (.3.8) and return to vector systems of conservation laws in Chapter 6. To begin, let a(u) = df(u) du (.3.9a) and write (.3.8) in the convective form u t + a(u)u =: (.3.9b) Regard u( t) as a function of twovariables and compute its total (directional) derivative du = u t dt + u d =(u t + d dt u )dt: (.3.)

14 4 Introduction Let us evaluate (.3.) in the direction given by Then, using (.3.9b) we see that d dt = a(u): (.3.) du =(u t + au )dt =: Thus, u( t) is constant along the curve d=dt = a. This curve is called a characteristic of the dierential equation (.3.8). Characteristics can be used to construct solutions of initial or initial boundary value problems. Let us begin with an initial value problem for (.3.8) on ; <<, t>, with the initial condition u( ) = () ; <<: (.3.2) Since u is constant along the characteristic curves, it must have the same value that it had initially. Thus, u = ( ) along the characteristic that passes through ( ). This characteristic satises the ordinary initial value problem d dt = a( ) a t> () = : (.3.3) Integrating, we see that the characteristic is the straight line = + a t: (.3.4) This procedure can be repeated to trace other characteristics and thereby construct the solution. The solution of (.3.8) is implicitly given by the formula u( t) =( ; a( )t) (.3.5) however, this formula is rather obscure. Let us clarify the situation with some eamples. Eample.3.3. The simplest case occurs when a is a constant and f(u) =au. All of the characteristics are parallel straight lines with slope =a. The solution of the initial value problem (.3.8,.3.2) is u( t) = ( ; at) and is, as shown in Figure.3., a wave that maintains its shape and travels with speed a.

15 .3. Hyperbolic Conservation Laws 5 t = + at a u(,t) at u(,) = φ() u(,t) = φ(-at) at Figure.3.: Characteristic curves and solution of the initial value problem (.3.8,.3.2) when a is a constant. t = + ( )t φ P φ φ = + ( )t φ Figure.3.2: Characteristic curves for two initial points and for Burgers' equation. The characteristics intersect at a point P.

16 6 Introduction Eample.3.4. Setting a(u) =u and f(u) =u 2 =2 in (.3.8,.3.9a) yields the inviscid Burgers' equation u t + 2 (u2 ) =: (.3.6) Again, consider an initial value problem having the initial condition (.3.2), so the characteristic is given by (.3.4) with a = u( ) = ( ), i.e., = + ( )t: (.3.7) The characteristics are straight lines, but they are not parallel as they were in Eample.3.3. Their slope depends on the value of the initial data thus, the characteristic passing through the point ( ) has slope =( ). The fact that the characteristics are not parallel introduces a diculty that was not present in the linear problem of Eample.3.3. Consider characteristics passing through ( ) and ( ) and suppose that ( ) > ( ) for >. Since the slope of the characteristic passing through ( ) is less than the slope of the one passing through ( ), the two characteristics will intersect at a point, say, P as shown in Figure.3.2. The solution would appear to be multivalued at points such as P. In order to help clarify matters, let's eamine the specic choice of given by La [4] () = 8 < : if < ; if < if : (.3.8) Using (.3.7), we see that the characteristic passing through the point ( ) satises = 8 < : + t if < +(; )t if < if : (.3.9) Several characteristics are shown in Figure.3.3. The characteristics rst intersect at t =. After that, the solution would presumably be multivalued, as shown in Figure.3.4. It's, of course, quite possible for multivalued solutions to eist however, (i) they are not observed in physical situations and (ii) they do not satisfy (.3.8) in any classical sense. Discontinuous solutions are often observed in nature once characteristics of the

17 .3. Hyperbolic Conservation Laws 7 corresponding conservation law model have intersected. They also do not satisfy (.3.8), but they might satisfy the integral form of the conservation law (.3.). We eamine the simplest case when two classical solutions satisfying (.3.8) are separated by a single smooth curve = (t) across which u( t) is discontinuous. For each t > we assume that <(t) <and let superscripts - and + denote conditions immediately to the left and right, respectively, of = (t). Then, using (.3.), we have d dt Z or, dierentiating the integrals Z ; Z ud = d ; dt [ ud + u t d + u ; _ ; + Z Z + ud] =;f(u)j + u td ; u + _ + = ;f(u)j : The solution on either side of the discontinuity was assumed to be smooth, so (.3.8) holds in ( ; )and( + ) and can be used to replace the integrals. Additionally, since is smooth, _ ; = _ + = _. Thus, we have or ;f(u)j ; + u ; _ ; f(u)j + ; u + _ = ;f(u)j _(u + ; u ; )=f(u + ) ; f(u ; ): (.3.2) Let [q] q + ; q ; (.3.2) denote the jump in a quantity q and write (.3.2) as [u] _ =[f(u)]: (.3.22) Equation (.3.22) is called the Rankine-Hugoniot jump condition and the discontinuity is called a shock wave. We can use the Rankine-Hugoniot condition to nd a discontinuous solution of Eample.3.4. Eample.3.5. For t<, the discontinuous solution of (.3.6,.3.8) is as given in Eample.3.4. For t, we hypothesize the eistence of a single shock wave, passing

18 8 Introduction t Figure.3.3: Characteristics for Burgers' equation (.3.6) with initial data given by (.3.8). u(,) u(,/2) 2 2 u(,) u(,3/2) 2 2 Figure.3.4: Multivalued solution of Burgers' equation (.3.6) with initial data given by (.3.8). The solution u( t) isshown as a function of for t =,/2,, and 3/2.

19 .3. Hyperbolic Conservation Laws 9 through ( ) in the ( t)-plane. As shown in Figure.3.5, the solution of Eample.3.4 can be used to infer that u ; = and u + =. Thus, f(u ; ) = (u ; ) 2 =2 = =2 and f(u + )=(u + ) 2 =2=. Using (.3.22), the velocity of the shock wave is _ = 2 : Integrating, we nd the shock location as = 2 t + c: Since the shock passes through ( ), the constant of integration c ==2, and = (t +): (.3.23) 2 The characteristics and shock wave are shown in Figure.3.5 and the solution u( t) is shown as a function of for several times in Figure.3.6. Let us consider another problem for Burgers' equation with dierent initial conditions that will illustrate another structure that arises in the solution of nonlinear hyperbolic systems. Eample.3.6. Consider Burgers' equation (.3.6) subject to the initial conditions () = 8 < : if < if < if : (.3.24) Using (.3.7) and (.3.24), we see that the characteristic passing through ( ) satises = 8 < : if < ( + t) if < + t if : (.3.25) These characteristics, shown in Figure.3.7, may be used to verify that the solution, shown in Figure.3.8, is continuous. nonlinear hyperbolic systems are discussed in La [4]. Additional considerations and diculties with Thus far, we have only considered initial value problems for (.3.8). Initial-boundary problems are also possible however, boundary conditions cannot be prescribed arbitrarily

20 2 Introduction t ξ = (t + )/2 Figure.3.5: Characteristics and shock discontinuity for Eample.3.5. u(,) u(,/2) 2 2 u(,) u(,3/2) 2 2 Figure.3.6: Solution u( t) of Eample.3.5 as a function of at t =, /2,, and 3/2. The solution is discontinuous for t>.

21 .3. Hyperbolic Conservation Laws 2 t u(,) Figure.3.7: Characteristics for Eample.3.6. u(,/2) 2 2 u(,) u(,3/2) 2 2 Figure.3.8: Solution u( t) of Eample.3.6 as a function of at t =, /2,, and 3/2.

22 22 Introduction t (,t ) = at (,t ) t - /a - a t Figure.3.9: Characteristics for the linear initial-boundary value problem (.3.26a). and are determined by the characteristic directions (.3.). Let us begin with the linear problem u t + au = << t> (.3.26a) u( ) = () : (.3.26b) Suppose that a >. As shown in Figure.3.9, the solution at a point ( t ) where > at is determined by the initial condition at ( ; at ). On the other hand, the solution at a point ( 2 t 2 ) where 2 < at 2 is not determined by the initial condition and requires a boundary value to be prescribed at ( t 2 ; 2 =a). Therefore, the proper boundary condition to apply is u( t)= (t) t>: (.3.26c) The solution of the initial-boundary value problem (.3.26a) is given by (t ; =a) if <<at u( t) = ( ; at) if at < : (.3.27) There are no boundary conditions at =. However, if a <, the situation would be reversed and boundary conditions would be needed at = and not at =. Determining the proper boundary conditions for nonlinear problems can be quite complicated. For eample, consider solving (.3.9b) on <<, t>witha(u( t)) >

23 .3. Hyperbolic Conservation Laws 23 and a(u( t)) <. As shown on the left of Figure.3., such a problem would need boundary conditions at both = and =. On the other hand, a problem with a(u( t)) < and a(u( t)) > (Figure.3., right) would not require any boundary conditions. t t Figure.3.: Characteristics for a nonlinear initial-boundary value problem with a(u( t)) > and a(u( t)) < (left) and with a(u( t)) < anda(u( t)) > (right). Problems. A Riemann problem is a Cauchy (initial value) problem with piecewise constant initial data. Consider the Riemann problem for the inviscid Burgers' equation u t + 2 (u2 ) = u( ) = ul if < u R if : where u L and u R are constants. Solve this problem in the two cases when u L >u R and when u L <u R. Sketch several characteristic curves and sketch the solution as a function of at a few times. As a hint, you may want to rst solve a problem with continuous initial data, e.g., u( ) = 8 < : u L ul ( ; )+ u R ( + ) 2 2 u R if <; if ; < if < and take the limit as!. You may also consult La [4].

24 24 Introduction 2. Use the method of characteristics to construct a solution of the inhomogeneous initial value problem u t + au = b() ; << t> u( ) = () ; << where a is aconstant and b() is smooth. 3. A scalar conservation law having cylindrical symmetry has the form d dt Z urdr = ;f(u)rj where r is a radial coordinate. If solutions are smooth this may be written as the rst-order hyperbolic partial dierential equation u t + r [f(u)r] r =: Solve an initial-boundary value problem for this equation on r >, t > when f(u) = au with a a positive constant. Assume the initial conditions prescribe u(r ) = (r) r> where (r) issmooth. What boundary conditions, if any, must be prescribed? 4. Consider a linear vector systems of m conservation laws of the form u t + Au = where A is a constant m m matri. Let P be the m m matri that reduces A to the canonical form P ; AP = : If A has m distinct real eigenvalues then is the diagonal matri = and the partial dierential system is called hyperbolic. (Additionally, if has m comple eigenvalues the system is called elliptic. These etend the notions of hyperbolicity and ellipticity thatwere introduced in Section.2. m 3 7 5

25 .3. Hyperbolic Conservation Laws Assume that the given partial dierential system is hyperbolic and introduce the transformation u = Pv: Show that v satises v t + v = : Thus, the transformation has uncoupled the system (ecept possibly for the initial and/or boundary conditions). It now has the scalar form (v i ) t + i (v i ) = i = 2 m: This system may be analyzed using characteristics in the same manner as for the scalar case Consider the wave equation written as a rst-order system of two equations with A = ;c ;c Find P,, and the diagonal system satised by v. Determine the characteristics and, hence, v. Knowing v, determine u( t) such that it satises the initial data u( ) = u () u 2() : :

26 26 Introduction

27 Bibliography [] W.E. Boyce and R.C. DiPrima (contibuter). Elementary Dierential Equations and Boundary Value Problems. John Wiley and Sons, New York, sith edition, 996. [2] P.R. Garabedian. Partial Dierential Equations. John Wiley and Sons, New York, 964. [3] D. Gottlieb and S.A. Orszag. Numerical Analysis of Spectral Methods: Theory and Applications. Regional Conference Series in Applied Mathematics, No. 26. SIAM, Philadelphia, 977. [4] P.D. La. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Regional Conference Series in Applied Mathematics, No.. SIAM, Philadelph, 973. [5] H.F. Weinberger. A First Course in Partial Dierential Equations. Ginn-Blaisdell, Waltham,