CENTRAL SCHEMES FOR THE MODIFIED BUCKLEY-LEVERETT EQUATION

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1 CENTRAL SCHEMES FOR THE MODIFIED BUCKLEY-LEVERETT EQUATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Ying Wang, B.Sc., M.Sc., M.A.S. Graduate Program in Mathematics The Ohio State University 2 Dissertation Committee: Professor Chiu-Yen Kao, Advisor Professor Avner Friedman Professor Fei-Ran Tian

2 c Copyright by Ying Wang July, 2

3 ABSTRACT In fluid dynamics, the Buckley-Leverett (BL equation is a transport equation used to model two-phase flow in porous media. by water-drive in oil reservoir simulation. One application is secondary recovery The modified Buckley-Leverett (MBL equation differs from the classical BL equation by including a balanced diffusivedispersive combination. The dispersive term is a third order mied derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profile for certain Riemann problems as suggested by the eperimental observations. In this thesis, we first show that the solution of the finite interval [, L] boundary value problem converges to that of the half-line [, + problem for the MBL equation as L +. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we etend the classical central schemes for the hyperbolic conservation laws to solve the MBL equation which is of pseudo-parabolic type. This etension can also be applied to other conservation law solvers. Numerical results confirm ii

4 the eistence of non-monotone water saturation profiles consisting of constant states separated by shocks, which is consistent to the eperimental observations. The two-dimensional physical space is a general setting for the underground oil recovery. In this thesis, we also include the derivation of the two-dimensional etension of the MBL equation. iii

5 To my teachers iv

6 ACKNOWLEDGMENTS I would first like to address special thanks to my advisor, Prof. Chiu-Yen Kao for her generous support, constant advice and endless patience in improving the results of this work, and more importantly, for setting a role model as a dedicating woman mathematician. I also want to thank my committee members, Profs. Avner Friedman and Fei-Ran Tian for their guidance and encouragements, not only during this thesis preparation process, but also throughout the years I stayed at OSU. It has been a long journey since the time I left home. What is happening now all started from an undergraduate scholarship I was awarded by Singapore Ministry of Education, without which, I would not have gone this far. It were the many Computer Scientists and Mathematicians I met at National University of Singapore (NUS who led me into the beautiful science world, among which, the lecture clips and conversation moments with Profs. Tay Yong Chiang and Andrew Lim still flash through my mind every now and then. Meanwhile then, the professors and graduate students I met at computational science department, NUS, motivated most of my early research work. This joint effort has later benefited my entire graduate school studies. I switched major of study from Computer Science to Mathematics when I started v

7 graduate school study at Georgia Tech. Amazingly enough, the Mathematics professors at Georgia Tech transformed me from one who could barely work out a calculus problem to a qualified Mathematics Ph.D. student. In addition to the Georgia Tech professors, I want to epress my gratitude to Profs. Douglas Ulmer at University of Arizona then and Weishi Liu at University of Kansas, who invited me for campus visits when I applied for the Ph.D. programs in their institutions. Their approval and recognition were more encouraging to me than they may have thought. Along the same line, I would like to thank Prof. Saleh Tanveer for having me here as a Mathematics Ph.D. student. I have benefited from the tutelage of several faculty and staff here at the Ohio State University. In particular, my appreciation goes to my recommendation letter writers. Thank you all for paving the way for my future development in academia. It is my deepest desire to dedicate this work to all my teachers, at every stage of my life. I am also indebted to all my friends. The friends I met in Singapore have painted my college life as colorful as it could be. Some are still very influential to date. The midnight pendulum conversation, the movie premieres, the Pandan valley days, and many more instances remind me the happy time. The friends I met in the United States have been good companion. They are the sweetest memory of my graduate school eperience. Especially, the honest criticisms have always helped me re-orient myself away from the myriad distractive matters. To all my friends, younger and older, local and remote, men and women, I am vi

8 deeply grateful to your support for all these years. The joys and tears we shared shaped my life to the full etent, both personally and academically. Last but not least, I want to thank my parents, for many reasons. The freedom they provided to me when I was little gave me infinite opportunities to eplore the world; the unconditional love they constantly epress to me held me through the rough days. I wish them well. vii

9 VITA 25 Present Ph.D. in Mathematics, The Ohio State University Master of Applied Statistics, The Ohio State University M.Sc. in Applied Mathematics, Georgia Institute of Technology Research Assistant, National University of Singapore B.Sc. with First Class Honors in Computer and Information Science, National University of Singapore B.Sc. (with Merit in Computer and Information Science, National University of Singapore. viii

10 FIELDS OF STUDY Major Field: Applied Mathematics Studies in Numerical Solutions of Hyperbolic Conservation Laws Numerical Analysis and Scientific Computing Application of Differential Equations i

11 TABLE OF CONTENTS Abstract Dedication Acknowledgments Vita ii iv v viii List of Figures ii List of Tables iv CHAPTER PAGE Introduction Underground Oil Production Process Classical Buckley-Leverett (BL Equation Derivation of the Modified Buckley-Leverett Equation.. 2 Half Line v.s. Finite Interval Domain Half Line Problem Finite Interval Boundary Value Problem Comparisons Definitions and Lemmas A Proposition Proof of Theorem Numerical Schemes Second-Order Schemes Trapezoid Scheme Midpoint Scheme

12 3.2 A Third Order Semi-discrete Scheme Computational Results Accuracy Analysis Traveling-Wave Solutions Parameter Regimes Eamples D Modified Buckley-Leverett Equation Derivation of the 2D MBL Equation Conclusion APPENDICES A Proof of the lemmas A. Proof to lemma A.2 Proof to lemma A.3 Proof to lemma A.4 Proof to lemma Bibliography i

13 LIST OF FIGURES FIGURE PAGE. Demonstration of secondary recovery during the underground oil production process (courtesy to MPG Petroleum, Inc. com/fundamentals.html The flu function and its derivative for Buckley-Leverett equation: (a u f(u = 2 ; (b f (u = 2Mu( u with M = u 2 +M( u 2 (u 2 +M( u (a: The triple-valued solution of BL; (b: Determination of the shock location using the equal area rule The entropy solution of the classical BL equation (M = 2, α = (a < ub =.7 α, the solution consists of one shock at = f(u B t u B ; (b α < ub =.98 <, the solution consists of a rarefaction between u B and α for f (u B < < f (α and a shock at t = f(α t α.5 Courtesy to [8]: Snapshots of the saturation profile versus depth for si different applied flues in initially dry 2/3 sand measured using light transmission. At the highest (.8 cm/min and lowest (7.9 4 cm/min flues the profiles are monotonic with distance and no saturation overshoot is observed, while all of the intermediate flues ehibit saturation overshoot Critical values of u when M = 2: α.86 and β The bifurcation diagram of the MBL equation (4. with the bifurcation parameters (τ, u B, where M = ii

14 4.3 Given a fied τ, the three qualitatively different solution profiles due to different values of u B. In particular, when τ > τ and u< u B < ū, the solution profiles (Figure 4.3(b displays non-monotonicity, which is consistent with the eperimental observations ([8]. Figures 4.3(a, 4.3(b and 4.3(c are demonstrative figures Numerical solutions to MBL equation with parameter settings fall in different regimes of the bifurcation diagram (Figure 4.2. The color coding is for different time: T (blue, 2T (green, 3 T (magenta and T (black. The results are discussed in eamples 6. In figures M 4.4(d 4.4(f, α = = 2 for M = M The numerical solutions of the MBL equation at T = with τ = and different u B values. The results are discussed in eample Numerical solutions to MBL equation with u B close to ū τ=5.98. The color coding is for different time: T (blue, 2 T (green, 3 T (magenta and T (black. The results are discussed in eample Numerical solutions to MBL equation with u B close to u τ=5.68. The color coding is for different time: T (blue, 2 T (green, 3 T (magenta and T (black. The results are discussed in eample Numerical solutions to MBL equation with small constant u B =.6 and different τ values. The figures on the second and third rows are the magnified versions of the first row at t = T and t = T respectively. 4 The color coding is for different time: T (blue, 2 T (green, 3 T (magenta and T (black. The results are discussed in eamples The numerical solutions of MBL equation at T =.5 with ɛ =. (blue, ɛ =.2 (yellow, ɛ =.3 (magenta, ɛ =.4 (green, and ɛ =.5 (black. The view windows are zoomed into the regions where different ɛ values impose different solution profiles. The results are discussed in eample iii

15 LIST OF TABLES TABLE PAGE 3. Flow chart for Trapezoid Scheme Flow chart for Midpoint Scheme The accuracy test for the trapezoid scheme for the MBL equation (4. with ɛ = and M = The accuracy test for the third order semi-discrete scheme for the MBL equation (4. with ɛ = and M = pairs of (τ, u B values with either fied τ value varying u B values or fied u B value varying τ values used in Eamples 6. Notice that M α = = 2 for M = M+ 3 iv

16 CHAPTER INTRODUCTION For the past fifth years, various research activities have been contacted to assist oilreservoir management. The main purpose is to provide an information database that can help the oil companies maimize the oil and gas recovery. Unfortunately, to obtain an accurate prediction of reservoir flow scenarios is a challenging task. One of the reasons is that we can never get a complete and concrete characterization of the rock parameters that influence the flow pattern. And even if we did, we would not be able to model the process that utilizes all available information, since this would require a tremendous amount of computer resources that eceed by far the capabilities of modern multi-processor computers. On the other hand, we do not need, nor do we seek a simultaneous description of the flow scenario on all scales down to the pore scale. For reservoir management it is usually sufficient to provide the general trends in the reservoir flow pattern. In this chapter, we will first provide some background knowledge of the underground oil production process in section.. In section.2, we will then introduce the classical Buckley-Leverett (BL equation, which models the water-saturation during secondary recovery by water-drive in oil reservoir simulation. Despite the simplicity, BL equation has its limitations. To improve the model in the sense of getting a

17 CHAPTER. INTRODUCTION saturation profile which is more consist to the eperimental observations, we consider the modified Buckley-Leverett (MBL equation. In section.3, we will show the derivation of MBL equation.. Underground Oil Production Process Initially, an oil reservoir is at an equilibrium stage, and contains oil/gas, and water, separated by gravity. This equilibrium has been established over millions of years with gravitational separation and geological and geothermal processes. When a well is drilled through the upper non-permeable layer and penetrates the upper oil cap, this equilibrium is immediately disturbed. Oil flows out of the reservoir due to overpressure. This in turn, sets up a flow inside the reservoir and oil flows towards the well, which in turn may induce gravitational instabilities. Also the capillary pressures will act as a (minor driving mechanism, resulting in local perturbations of the situation. During the above process, perhaps 2 percent of the oil present is produced until a new equilibrium is achieved. This process is called primary production by natural drives. Notice that a sudden drop in pressure also may have numerous other intrinsic effects. As pressure drops, less oil/gas is flowing, and eventually the production is no longer economically sustainable. Then the operating company may start secondary production by engineered drives. These are processes based on injecting water or gas into the reservoir (see Figure.. The reason for doing this is two folds: some of the pressure is rebuilt or even increased, and secondly one tries to push out more profitable oil 2

18 CHAPTER. INTRODUCTION Figure.: Demonstration of secondary recovery during the underground oil production process (courtesy to MPG Petroleum, Inc. com/fundamentals.html. with the injected substance. One may perhaps produce another 2 percent of the oil by such processes and engineered drives are standard procedure at most locations. This process is called secondary recovery. In order to produce even more oil, Enhanced Oil Recovery (EOR, or tertiary recovery techniques may be employed. Among these are heating the reservoir or injection of sophisticated substances like foam, polymers or solvents. Polymers are supposed to change the flow properties of water, and thereby to more efficiently push out oil. Similarly, solvents change the flow properties of the oil, for instance by developing miscibility with an injected gas. In some sense, one tries to wash the pore walls for most of the remaining oil. The other technique is based on injecting steam, which will heat the rock matri, and thereby, hopefully, change the flow properties of the oil. At present, such EOR techniques are considered too epensive for large-scale commercial 3

19 CHAPTER. INTRODUCTION use, but several studies have been conducted and the mathematical foundations are being carefully investigated, and at smaller scales EOR is being performed. Here, the terms primary, secondary, and tertiary are ambiguous. EOR techniques may be applied during primary production, and secondary recovery may be performed from the first day of production. In this thesis, we focus on the mathematical modeling and numerical simulations of the secondary recovery process. For the ease of modeling, we start with onedimensional model describing the horizontal flow..2 Classical Buckley-Leverett (BL Equation The classical Buckley-Leverett (BL equation [3] is a simple and effective model for two-phase fluid flow in a porous medium. One application is secondary recovery by water-drive in oil reservoir simulation. In this case, the two phases are oil and water, and the flow takes place in a porous medium of rock or sand. In one space dimension the equation has the standard conservation form u t + (f(u = in Q = {(, t : >, t > } (. u(, = (, u(, t = u B t [, with the flu function f(u being defined as u <, f(u = u 2 u, u 2 +M( u 2 u >. (.2 4

20 CHAPTER. INTRODUCTION In this content, u : Q [, ] denotes the water saturation, and so lies between and (e.g. u = means pure water, and u = means pure oil, u B is a constant which indicates water saturation at =, and M > is a constant representing the water/oil viscosity ratio. The classical BL equation (. is a prototype for conservation laws with conve-concave flu functions. The graph of f(u and f (u with M = 2 is given in Figure.2. Notice that in Figure.2 (a, at u = α, the tangent line coincide with the secant line connecting (α, f(α and (, f(. We will discuss the meaning of this point in the later part of this section. f(u.5.5 α u (a f (u u (b Figure.2: The flu function and its derivative for Buckley-Leverett equation: (a u f(u = 2 ; (b f (u = 2Mu( u with M = 2. u 2 +M( u 2 (u 2 +M( u 2 2 Due to the possibility of the eistence of shocks in the solution of the hyperbolic 5

21 CHAPTER. INTRODUCTION conservation laws (., the weak solutions are sought. The function u L (Q is called a weak solution of the conservation laws (. if Q { u φ } t + f(u φ = for all φ C (Q. Notice that the weak solution is not unique. Among the weak solutions, the entropy solution is physically relevant and unique. The weak solution that satisfies the Oleinik entropy condition [8] f(u f(u l u u l s f(u f(u r u u r for all u between u l and u r (.3 is the entropy solution, where u l, u r are the function values to the left and right of the shock respectively, and the shock speed s satisfies the Rankine-Hugoniot jump condition [6, ] s = f(u l f(u r u l u r. (.4 Consider the Riemann problem with initial condition u B if = u(, = if >, by following the characteristics, we can construct the triple-valued solution as shown in Figure.3(a. Notice that the characteristic velocities are f (u so that the profile at time t is simply the graph of tf (u turned sideways [4]. We can use the equal area rule: the shaded area on the left of the shock = the shaded area on the right of the shock 6

22 CHAPTER. INTRODUCTION u B u B α f (u t f (u t u (a (b Figure.3: (a: The triple-valued solution of BL; (b: Determination of the shock location using the equal area rule. as shown in Figure.3(b, i.e. α u tf (u du (α u = u u tf (u du, t(f(α f(u α = t(f(u f(, = f(α α t. Furthermore, from the triple-valued solution, we have that = tf (α, therefore, the post-shock value α satisfies f (α = f(α α, (.5 7

23 CHAPTER. INTRODUCTION which gives that α = M M +. (.6 The location of α and the special property (.5 that α satisfies are shown in Figure.2 (a for M = 2. Notice that in this case, u r = and f(u r =, hence the Rankine-Hugoniot condition (.4 is automatically satisfies. Hence, replacing the triple-valued solution by a shock located at = f(α α t gives the entropy solution. And the entropy solution of the classical BL equation can be classified into two categories:. If < u B α, the entropy solution has a single shock at t = f(u B u B. 2. If α < u B <, the entropy solution contains a rarefaction between u B and α for f (u B < t < f (α and a shock at t = f(α α. These two types of solutions are shown in Figure.4 for M = 2. In either case, the entropy solution of the classical BL equation (. is a non-increasing function of at any given time t >. However, the eperiments [8] of two-phase flow in porous medium reveal comple infiltration profiles, which may involve overshoot, i.e., profiles may not be monotone as given in Figure.5. This suggests the need of modification to the classical BL equation (.. 8

24 CHAPTER. INTRODUCTION ub =.7 ub =.98 u.5 u.5.5 t (a.5 t (b 2 Figure.4: The entropy solution of the classical BL equation (M = 2, α = (a < ub =.7 α, the solution consists of one shock at = f(u B t u B ; (b α < ub =.98 <, the solution consists of a rarefaction between u B and α for f (u B < < f (α and a shock at = f(α. t t α 9

25 CHAPTER. INTRODUCTION Figure.5: Courtesy to [8]: Snapshots of the saturation profile versus depth for si different applied flues in initially dry 2/3 sand measured using light transmission. At the highest (.8 cm/min and lowest (7.9 4 cm/min flues the profiles are monotonic with distance and no saturation overshoot is observed, while all of the intermediate flues ehibit saturation overshoot.

26 CHAPTER. INTRODUCTION.3 Derivation of the Modified Buckley-Leverett Equation To better understand the problem, we go back to the origins of (.. Let S i = saturation of oil/water i = o,w where S o + S w = (.7 that is, the medium is assumed to be completely saturated. The balance of mass yields where d 2 φs i (, t d = q i ( q i ( 2 for any, 2, and i =, w, dt φ = porosity of the medium = relative volume occupied by the pores, so lies between and, q i = discharge of oil/water. Since, 2 are arbitrary, if S i and q i are smooth, we have the differential form of the conservation law φ S i t + q i =. (.8 Notice that due to the complete saturation assumption (.7, we have that q o + q w = q = const in space. (.9

27 CHAPTER. INTRODUCTION Throughout of this thesis, we consider it constant in time as well. The discharge of each phase is modeled by Darcy s law [2] where For the ease of notation, we denote then (. becomes q i = k k ri(s i P i, i = o, w. (. µ i k = absolute permeability, k ri = relative permeability, µ i = viscosity, P i = phase pressure. λ i = k k r i (S i µ i, (. q i = λ i P i. (.2 Instead of considering constant capillary pressure as adopted by the classical BL equation (., Hassanizadeh and Gray [9][] have defined the dynamic capillary pressure as P c = P o P w = p c (S w φτ S w t (.3 where p c (S w is the static capillary pressure and τ is a positive constant, and Sw t the dynamic effects. To simplify the notation, let s write (.3 as is P o P w = r.h.s. 2

28 CHAPTER. INTRODUCTION where r.h.s. = p c (S w φτ S w t, (.4 then P o P w = [r.h.s.]. Combine (.2, we get q o q w = [r.h.s.]. (.5 λ o λ w Notice that (.9 gives q o λ o q w λ w = q q w λ o q w λ w = q λ o λ o + λ w λ o λ w q w, and hence (.5 becomes q λ o + λ w q w = λ o λ o λ w [r.h.s.], and q w = λ w λ o + λ w q λ oλ w λ o + λ w [r.h.s.]. (.6 Plugging (.6 into the governing equation (.8 for S w, we get that φ S w + [ λ w q λ ] oλ w t λ o + λ w λ o + λ w [r.h.s.] =, and hence S w t + [ ] λ w q λ o + λ w φ = [ ] λ o λ w φ(λ o + λ w [r.h.s.]. 3

29 CHAPTER. INTRODUCTION By the definition of λ w, λ o, and r.h.s. in (. and (.4 respectively, we have that S w + [ ] kk rw (S w /µ w q t kk ro (S o /µ o kk rw (S w /µ w φ = [ ( ( kkro (S o /µ o ( kk rw (S w /µ w pc (S w τ S ] w. kk ro (S o /µ o kk rw (S w /µ w φ t Using Corey [6, 9] epressions, k rw (S w = S 2 w, k ro (S o = S 2 o, and combining with (.7, we get [ ] S w + Sw 2 q t Sw 2 + µw µ o ( S w 2 φ = [ ( k( S w 2 Sw 2 pc (S w µ w ( S w 2 + µ o Sw 2 φ If we rescale τ S ] w. t φ q and let u = S w = saturation of water, we get u t + [ ] u 2 u 2 + M( u 2 where = [ ( φ 2 k( u 2 u 2 pc (u q 2 µ w ( u 2 + µ o u 2 φ τ u ], t M = µ w µ o. 4 (.7

30 CHAPTER. INTRODUCTION Equation (.7 can be written as a more general form where u t + f(u = { H(u ( J(u τ u }. (.8 t f(u = u 2 u 2 + M( u 2 is the BL flu. In this thesis, we consider the linearized right hand side. In (.8 if we take H(u = ɛ 2, J(u = u ɛ, or equivalently if we assume porosity φ is small in (.7 by taking φ 2 q 2 k( u 2 u 2 µ w ( u 2 + µ o u 2 = ɛ2, p c (u φ = u ɛ, then the modified Buckley-Leverett equation (MBL is obtained u t + f(u = ɛ 2 u 2 + ɛ2 τ 2 u 2 t. (.9 Note that, if P c in (.3 is taken to be constant, then (.7 gives the classical BL equation; while if the dispersive parameter τ is taken to be zero, or equivalently dynamic effect in the pressure difference between the two phases is neglected, then (.9 gives the viscous BL equation, which still displays monotone water saturation profile. Thus, in addition to the classical second order viscous term ɛu, the MBL 5

31 CHAPTER. INTRODUCTION equation (.9 is an etension involving a third order mied derivative term ɛ 2 τu t. Van Dujin et al. [2] showed that the value τ is critical in determining the type of the profile. In particular, for certain Riemann problems, the solution profile of (.9 is not monotone when τ is larger than the threshold value τ, where τ was numerically determined to be.6 [2]. The non-monotonicity of the solution profile is consistent with the eperimental observations [8] as given in Figure.5. The classical BL equation (. is hyperbolic, and the numerical schemes for hyperbolic equations have been well developed (e.g. [4, 5, 4, 5, 7, 2]. The MBL equation (.9, however, is pseudo-parabolic, we will illustrate how to etend the central schemes [7, 2] to solve (.9 numerically. Unlike the finite domain of dependence for the classical BL equation (., the domain of dependence for the MBL equation (.9 is infinite. This naturally raises the question for the choice of computational domain. To answer this question, we will first study the MBL equation equipped with two types of domains and corresponding boundary conditions. One is the half line problem u t + (f(u = ɛu + ɛ 2 τu t in Q = {(, t : >, t > } u(, t = g u (t, u(, = u ( [, lim u(, t = t [, (.2 u ( = g u ( compatibility condition 6

32 CHAPTER. INTRODUCTION and the other one is the finite interval boundary value problem v t + (f(v = ɛv + ɛ 2 τv t in Q = {(, t : (, L, t > } v(, = v ( [, L] v(, t = g v (t, v(l, t = h(t t [, (.2 v ( = g v (, v (L = h( compatibility condition. Considering v ( for [, L] u ( = for [L, +, g u (t = g v (t g(t, h(t,(.22 we will show the relation between the solutions of problems (.2 and (.2. To the best knowledge of the author, there is no such study for MBL equation (.9. Similar questions were answered for BBM equation [, 2]. The organization of this thesis is as follows. Chapter 2 will bring forward the eact theory comparing the solutions of (.2 and (.2. The difference between the solutions of these two types of problems decays eponentially with respect to the length of the interval L for practically interesting initial profiles. This provides a theoretical justification for the choice of the computational domain. In chapter 3, high order central schemes will be developed for MBL equation in finite interval domain. We provide a detailed derivation on how to etend the central schemes [7, 2] for conservation laws to solve the MBL equation (.9. The idea of adopting numerical schemes originally designed for hyperbolic equations to pseudo-parabolic equations is not restricted to central type schemes only ([22, 23]. The numerical results in chapter 4 show that the water saturation profile strongly depends on the dispersive 7

33 CHAPTER. INTRODUCTION parameter τ value as studied in [2]. For τ > τ, the MBL equation (.9 gives nonmonotone water saturation profiles for certain Riemann problems as suggested by eperimental observations [8]. In chapter 5, we show the two-dimensional etension of MBL equation and discuss the preliminary numerical solutions. Chapter 6 gives the conclusion of the paper and the possible future directions. 8

34 CHAPTER 2 THE HALF LINE PROBLEM VERSUS THE FINITE INTERVAL BOUNDARY VALUE PROBLEM Let u(, t be the solution to the half line problem (.2, and let v(, t be the solution to the finite interval boundary value problem (.2. We consider the natural assumptions (.22. The eistence of solution for both the half line problem (.2 the finite interval boundary value problem (.2 can be proven similar to that in []. We will therefore omit that proof. The goal of this chapter is to develop an estimate of the difference between u and v on the spatial interval [, L] at a given finite time t. The main result of this section is Theorem 2.. (The main Theorem. If u ( satisfies C u [, L ] u ( = (2. > L where L < L and C u, are positive constants, then u(, t v(, t H L,ɛ,τ D ;ɛ,τ (te λl + D2;ɛ,τ (te λ(l L for some < λ <, D ;ɛ,τ (t > and D 2;ɛ,τ (t >, where L Y (, t H L,ɛ,τ := Y (, t 2 + (Y (, t 2 d. 9

35 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Notice that the initial condition (2. we considered is the Riemann problem. Theorem 2.. shows that the solution to the half line problem (.2 can be approimated as accurately as one wants by the solution to the finite interval boundary value prob- λl lem (.2 in the sense that D ;ɛ,τ (t, D 2;ɛ,τ (t, ɛ and λ(l L τ can be controlled. To prove theorem 2.., we first derive the implicit solution formulae for the half line problem and the finite interval boundary value problem in section 2. and section 2.2 respectively. The implicit solution formulae are in integral form, which are derived by separating the -derivative from the t-derivative, and formally solving a first order linear ODE in t and a second order non-homogeneous ODE in. In section 2.3, we use Gronwall s inequality multiple times to obtain the desired result in theorem Half Line Problem In this section, we derive the implicit solution formula for the half line problem (.2. For ease of reference, the equation is repeated here. u t + (f(u = ɛu + ɛ 2 τu t in Q = {(, t : >, t > } u(, t = g u (t = g(t, u(, = u ( [, lim u(, t = t [, (2.2 u ( = g u ( = g( compatibility condition. 2

36 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN To solve (2.2, we first rewrite (2.2 by separating the -derivative from the t- derivative, (I ɛ 2 τ 2 2 ( u t + ɛτ u = ɛτ u (f(u. (2.3 Notice that (2.3 can be viewed as a first order linear ODE in t. By multiplying integrating factor, we get (I ɛ 2 τ 2 ( e t ɛτ ut + e t ɛτ (I 2 ɛ 2 τ 2 2 ( ɛτ u = ɛτ u (f(u ( = t ɛτ u (f(u ( ue t ɛτ We formally integrate (2.4 over [, t] to obtain (I ɛ 2 τ 2 (I ɛ 2 τ 2 Furthermore, we let 2 2 ( ue t ɛτ u ( u e t ɛτ u = = t t ( ɛτ u (f(u ( ɛτ u (f(u e t ɛτ, e t ɛτ. (2.4 e s ɛτ ds, e t s ɛτ ds. (2.5 A = u e t ɛτ u, (2.6 then (2.5 can be written as A ɛ 2 τa = t ( ɛτ u (f(u e t s ɛτ ds where =, i.e. A ɛ 2 τ A = t ( ɛ 3 τ u + 2 ɛ 2 τ (f(u e t s ɛτ ds. (2.7 2

37 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Notice that (2.7 is a second-order non-homogeneous ODE in -variable along with the boundary conditions A(, t = u(, t e t ɛτ u ( = g(t e t ɛτ g(, (2.8 A(, t = u(, t e t ɛτ u ( =. To solve (2.7, we first solve the corresponding linear homogeneous equation with the non-zero boundary conditions (2.8. (A h ɛ 2 τ A h =, A h (, t = g(t e t ɛτ g(, A h (, t =, and the solution is A h (, t = (g(t e t ɛτ g(e. We then find a particular solution for the non-homogeneous equation with zero boundary conditions B ɛ 2 τ B = t ( ɛ 3 τ u + 2 ɛ 2 τ (f(u e t s ɛτ d, (2.9 B(, t =, B(, t =. 22

38 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN We break the right hand side of (2.9 into two parts and consider them separately. Let B satisfy B ɛ 2 τ B = t B (, t =, B (, t =. t s ɛ 3 τ 2 ue ɛτ d, We look for the Green s function G(, ξ ( ξ (, for B that satisfies G G = δ( ξ, ɛ 2 τ G(, ξ =, G(, ξ =, G(ξ, ξ = G(ξ +, ξ, G (ξ +, ξ G (ξ, ξ =, where =. Such a Green s function G(, ξ is G(, ξ = ɛ τ 2 (e +ξ e ξ. Hence, B (, t = ɛ 3 τ 2 t + = 2ɛ 2 τ τ t + G(, ξu(ξ, se t s ɛτ (e +ξ dξ ds e ξ u(ξ, se t s ɛτ dξ ds. 23

39 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Similarly, let B 2 satisfy B 2 t ɛ 2 τ B 2 = ɛ 2 τ (f(u e t s ɛτ ds, B 2 (, t =, B 2 (, t =. Since = + + [ ɛ τ = = 2 (f(u ξ G(, ξ dξ 2 (f(u ξ (e +ξ ( 2 f(u e +ξ f(u (e +ξ f(u e ξ dξ ] ξ=+ e ξ ( let Kernel K(, ξ where ξ (, be K(, ξ = 2 ξ= ɛ +ξ τ e ɛ sgn( ξe τ dξ, + sgn( ξe ξ (e +ξ + sgn( ξe ξ ξ dξ and hence B 2 (, t is B 2 (, t = ɛ 2 τ = 2ɛ 2 τ t + t + K(, ξf(ue t s ɛτ (e +ξ dξ ds + sgn( ξe ξ f(ue t s ɛτ dξ ds. 24

40 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Therefore, the solution for (2.9 is Therefore, B(, t =B (, t + B 2 (, t = 2ɛ 2 τ τ + 2ɛ 2 τ t + t + (e +ξ (e +ξ e ξ u(ξ, se t s ɛτ dξ ds + sgn( ξe ξ f(ue t s ɛτ dξ ds. A(, t = B(, t + A h (, t ( = B(, t + g(t e t ɛτ g( e. By (2.6, we get the implicit solution formulae for u(, t u(, t =A(, t + e t ɛτ u ( ( =B(, t + g(t e t ɛτ g( t + (e +ξ = 2ɛ 2 τ τ + 2ɛ 2 τ + t + (e +ξ ( g(t e t ɛτ g( e e e ξ + e t ɛτ u ( u(ξ, se t s ɛτ dξ ds + sgn( ξe ξ f(ue t s ɛτ dξ ds + e t ɛτ u (. ( Finite Interval Boundary Value Problem In this section, we derive the implicit solution formula for the finite interval boundary value problem (.2. For ease of reference, the finite interval boundary value problem 25

41 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN is repeated here. v t + (f(v = ɛv + ɛ 2 τv t in Q = {(, t : (, L, t > } v(, = v ( [, L] v(, t = g v (t = g(t v(l, t = h(t t [, (2. v ( = g v ( = g( v (L = h( compatibility condition. The idea of solving (2. is the same as that of solving (2.2. The only difference is that the additional boundary condition h(t at = L in (2. gives different boundary conditions for the non-homogeneous ODE in -variable. To solve for the solution of (2., we first rewrite (2. by separating the -derivative from the t-derivative, i.e., (I ɛ 2 τ 2 2 ( v t + ɛτ v = ɛτ v (f(v (2.2 and then multiply the integrating factor ( ( (I ɛ 2 τ 2 e t 2 ɛτ vt + e t ɛτ ɛτ v = ɛτ v (f(v ( ( (I ɛ 2 τ 2 ve t 2 ɛτ = t ɛτ v (f(v Integrate (2.3 over [, t], we get ( (I ɛ 2 τ 2 ve t 2 ɛτ v = ( (I ɛ 2 τ 2 v e t 2 ɛτ v = Denote t t ( ɛτ v (f(v ( ɛτ v (f(v e t ɛτ, e t ɛτ. (2.3 e s ɛτ ds, e t s ɛτ ds. A L = v e t ɛτ v, (2.4 26

42 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN then A L satisfies A L ɛ 2 τ(a L = t ( (f(v + ɛτ v e t s ɛτ ds where = (2.5 i.e., (A L ɛ 2 τ AL = t ( ɛ 3 τ v + 2 ɛ 2 τ (f(v e t s ɛτ d. (2.6 The non-homogeneous second order ODE (2.6 has the following boundary conditions A L (, t = v(, t e t ɛτ v ( = g(t e t ɛτ g(, A L (L, t = v(l, t e t ɛτ v (L = h(t e t ɛτ h(. The solution to the corresponding homogeneous equation with the boundary conditions (A L h ɛ 2 τ AL h = A L h(, t = g(t e t ɛτ g( A L h(l, t = h(t e t ɛτ h( is A L h = c (tφ ( + c 2 (tφ 2 (, 27

43 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN where c (t = g(t e t ɛτ g(, (2.7 c 2 (t = h(t e t ɛτ h(, (2.8 φ ( = e L e L φ 2 ( = e e L L+ e e L e e L, (2.9. (2.2 Now, we solve the corresponding non-homogeneous equation with zero boundary conditions: (B L ɛ 2 τ BL = t ( ɛ 3 τ v + 2 ɛ 2 τ (f(v e t s ɛτ d, (2.2 B L (, t =, B L (L, t =. We break the right hand side of (2.2 into two parts and consider them separately. Let B L satisfy (B L ɛ 2 τ BL = t B L (, t =, B L (L, t =. t s ɛ 3 τ 2 ve ɛτ d, 28

44 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN We look for the Green s function G L (, ξ, where ξ (, L, for B L that satisfies (G L ɛ 2 τ GL = δ( ξ, G L (, ξ =, G L (L, ξ =, G L (ξ, ξ = G L (ξ +, ξ, (G L (ξ +, ξ (G L (ξ, ξ =, where =. Such a Green s function G L (, ξ is G L (, ξ = ( 2(e 2L e +ξ 2L (+ξ + e e ξ e 2L ξ. Hence, B L (, t = ɛ 3 τ 2 t L = 2ɛ 2 τ τ(e 2L (e +ξ G L (, ξv(ξ, se t s ɛτ t L 2L (+ξ + e e ξ dξ ds e 2L ξ v(ξ, se t s ɛτ dξ ds. Similarly, let B L 2 satisfy (B2 L t ɛ 2 τ BL 2 = ɛ 2 τ (f(v e t s ɛτ ds, B L 2 (, t =, B L 2 (L, t =. 29

45 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Since = = L L [ = (f(v ξ G L (, ξ dξ ( (f(v 2(e 2L ξ 2(e 2L L f(v 2(e 2L + 2(e 2L (e +ξ f(v e +ξ 2L (+ξ + e 2L (+ξ + e e ξ e ξ ( e +ξ e 2L (+ξ ξ sgn( ξe L f(v (e +ξ +sgn( ξe ξ 2L (+ξ e e 2L ξ dξ ] ξ=l e 2L ξ 2L ξ sgn( ξe sgn( ξe 2L ξ dξ, ξ= dξ let K L (, ξ where ξ (, L be K L ( (, ξ = 2(e 2L and hence B L 2 (, t is e +ξ 2L (+ξ e + sgn( ξe ξ sgn( ξe 2L ξ, B L 2 (, t = ɛ 2 τ = (e +ξ t L 2ɛ 2 τ(e 2L 2L (+ξ e K L (, ξf(ve t s ɛτ t + dξ ds f(ve t s ɛτ + sgn( ξe ξ sgn( ξe 2L ξ dξ ds. 3

46 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Therefore, the solution for (2.2 is B L (, t =B L (, t + B L 2 (, t = 2ɛ 2 τ τ(e 2L 2ɛ 2 τ(e 2L (e +ξ t L t L 2L (+ξ e (e +ξ f(ve t s ɛτ 2L (+ξ + e + sgn( ξe ξ e ξ v(ξ, se t s ɛτ dξ ds e 2L ξ sgn( ξe 2L ξ dξ ds. Therefore, A L (, t = B L (, t + A L h(, t = B L (, t + ( g(t e t ɛτ g( φ ( + ( h(t e t ɛτ h( φ 2 (. By (2.4, we get the implicit solution formulae for v(, t v(, t =A L (, t + e t ɛτ v ( =B L (, t + = ( g(t e t ɛτ g( φ ( + 2ɛ 2 τ τ(e 2L 2ɛ 2 τ(e 2L + (e +ξ t L t L 2L (+ξ e ( g(t e t ɛτ g( φ ( + (e +ξ f(ve t s ɛτ ( h(t e t ɛτ h( φ 2 ( + e t ɛτ v ( 2L (+ξ + e + sgn( ξe ξ e ξ v(ξ, se t s ɛτ dξ ds e 2L ξ sgn( ξe 2L ξ dξ ds ( h(t e t ɛτ h( φ 2 ( + e t ɛτ v (. 3

47 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN 2.3 Comparisons With the implicit solution formulae for the half line problem and the finite interval boundary value problem derived in sections 2. and 2.2 respectively, we will prove in this section that the solution u(, t to the half line problem can be approimated as accurately as one wants by the solution v(, t to the finite interval boundary value problem as stated in Theorem 2... The idea of the proof is to decompose u(, t (v(, t respectively into two parts: U(, t and u L (, t (V (, t and v L (, t respectively. u L (, t (v L (, t respectively consists of terms involving the initial condition u ( (v ( respectively and the boundary conditions g(t (g(t and h(t respectively for the governing equation (2.2((2. respectively. U(, t (V (, t respectively enjoys zero initial condition and boundary conditions while satisfying a slightly different equation than (2.2((2. respectively. We estimate the difference between u(, t and v(, t by estimating the differences between u L (, t and v L (, t, U(, t and V (, t, then applying the triangle inequality. In section 2.3., we will give the definitions of the decomposition of u(, t (v(, t respectively and a list of lemmas that will be used in the proof of Theorem 2... The proof of the lemmas can be found in the appendi A. In addition, the norm H L,ɛ,τ to be used in Theorem 2.. will also be introduced in section In section 2.3.2, we will prove a critical estimate which is essential to the proof of Theorem 2... In section 2.3.3, we will give the maimum difference u L (, t v L (, t, and use it to derive u L (, t v L (, t H and U(, t V (, t L,ɛ,τ H by using Gronwall s L,ɛ,τ 32

48 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN inequality. In the end, the ultimate difference u(, t v(, t H L,ɛ,τ will be derived based on the triangle inequality Definitions and Lemmas To assist the proof of Theorem 2.. in section 2.3.3, we introduce some new notations in this section. We first decompose u(, t as sum of two terms U(, t and u L (, t, such that u(, t = U(, t + u L (, t [, + where u L = e t ɛτ u ( + c (te + ( u(l, t c (te L e t ɛτ u (L φ 2 ( (2.22 and c (t and φ 2 ( are given in (2.7 and (2.2 respectively. With this definition, u L takes care of the initial condition u ( and boundary conditions g(t at = and = L for u(, t. Then U satisfies an equation slightly different from the equation u satisfies in (2.2: U t ɛu ɛ 2 τu t = ( u t ɛu ɛ 2 τu t ( (ul t ɛ(u L ɛ 2 τ(u L t = (f(u + ɛτ u L(, t. (

49 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN In addition, U(, t has zero initial condition and boundary conditions at = and = L, i.e., U(, =, U(, t =, (2.24 U(L, t =. Similarly, for v(, t, let where v(, t = V (, t + v L (, t [, L] v L = e t ɛτ v ( + c (tφ ( + c 2 (tφ 2 ( (2.25 and c (t, c 2 (t and φ (, φ 2 ( are given in (2.7,2.8 and (2.9,2.2 respectively. With this definition, v L takes care of the initial condition v ( and boundary conditions g(t and h(t at = and = L for v(, t. Then V satisfies an equation slightly different from the equation v satisfies in (2.: V t ɛv ɛ 2 τv t = ( ( v t ɛv ɛ 2 τv t (vl t ɛ(v L ɛ 2 τ(v L t (2.26 = (f(v + ɛτ v L(, t. In addition, V (, t has zero initial condition and boundary conditions at = and = L, i.e., V (, =, V (, t =, (2.27 V (L, t =. 34

50 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Since, in the end, we want to study the difference between U(, t and V (, t, we define W (, t = V (, t U(, t for [, L]. Because of (2.23 and (2.26, we have W t ɛw ɛ 2 τw t = (f(v f(u + ɛτ (v L u L. (2.28 In lieu of (2.24 and (2.27, W (, t also has zero initial condition and boundary conditions at = and = L, i.e., W (, =, W (, t =, (2.29 W (L, t =. Now, in order to estimate u v, we can estimate W = V U and estimate u L v L separately. These estimates are done in section Net, we state the lemmas needed in the proof of Theorem 2... The proof of the lemmas can be found in the appendi A. In all the lemmas, we assume < λ < and u ( satisfies C u [, L ] u ( = (2.3 > L where L < L and C u are positive constants. Notice that the constraint λ (, is crucial in Lemmmas and Lemma f(u = u 2 u 2 +M( u 2 Du where D = f(α and α = α 35 M. M+

51 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Lemma (i + (ii + (iii + e +ξ e +ξ e ξ e ξ Lemma (i + (ii + (iii + e +ξ e +ξ Lemma e +ξ e ξ e λ ξ dξ e( λ. e λ λξ dξ 2ɛ τ λ 2. e λ u (ξ dξ 2C u e λl e +ξ + sgn( ξe ξ + sgn( ξe ξ (i φ ( e (ii φ 2 ( for [, L]. + sgn( ξe ξ. e λ λξ e λ ξ dξ ɛ τ + e( λ. dξ 2ɛ τ λ 2. e λ u (ξ dξ 2C u e λl = e L φ2 (.. (iii φ 2( 2 if ɛ for [, L]. Last but not least, the norm that we will use in Theorem 2.. and its proof is L Y (, t H L,ɛ,τ := Y (, t 2 + (Y (, t 2 d. ( A Proposition In this section, we will give a critical estimate, which is essential in the calculation of maimum difference u L (, t v L (, t in section By comparing u L (, t and v L (, t given in (2.22 and (2.25 respectively, it is clear that the coefficient u(l, t c (te L e t ɛτ u (L for φ 2 ( appeared in (2.22 needs to be compared 36

52 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN with the corresponding coefficient c 2 (t for φ 2 ( appeared in (2.25. In this section, we will find a bound for u(l, t c (te L u(l, t c (te L e t ɛτ u (L a τ (te bτ t e t ɛτ u (L as follows ɛτ e λl t + cτ ɛτ e (bτ t ɛτ e λ(l L (2.32 for some parameter-dependent constants a τ, b τ and c τ. The idea of the proof is to define a space-dependent function U c2 (, t = u(, t c (te e t ɛτ u ( (2.33 and show that U c2 (, t decays eponentially with respect to by using Gronwall s inequality and then evaluate U c2 at = L to obtain (2.32. Based on the implicit solution formula (2. derived in section 2., we have U c2 (, t = 2ɛ 2 τ τ + 2ɛ 2 τ t + t + (e +ξ (e +ξ and based on the relationship between U c2 [ t U c2 (, t = 2ɛ 2 τ τ ɛ 2 τ t + t + t + + (e +ξ (e +ξ (e +ξ e ξ u(ξ, se t s ɛτ dξ ds + sgn( ξe ξ f(ue t s ɛτ dξ ds, (e +ξ and u given in (2.33, we have e ξ U c2 (ξ, se t s ɛτ dξ ds e ξ c (se ξ e t s ɛτ dξ ds ] e ξ u (ξe s ɛτ e t s ɛτ dξ ds + sgn( ξe ξ f(ue t s ɛτ dξ ds, 37

53 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN and based on Lemma 2.3., we can get an inequality in terms of U c2 [ t + U c2 (, t 2ɛ 2 τ e +ξ ɛ τ e ξ U c2 (ξ, s e t s ɛτ dξ ds τ t + + e +ξ ɛ τ e ξ c (s e ξ e t s ɛτ dξ ds t + ] + e +ξ ɛ τ e ξ u (ξ e t ɛτ dξ ds + D [ t + (2.34 e +ξ 2ɛ 2 ɛ τ + sgn( ξe ξ U c2 (ξ, s e t s ɛτ dξ ds τ t + + e +ξ ɛ τ + sgn( ξe ξ c (s e ξ e t s ɛτ dξ ds t + ] + e +ξ ɛ τ + sgn( ξe ξ u (ξ e t ɛτ dξ ds. To show that U c2 (, t decays eponentially with respect to, we first pull out an eponential term by writing U c2 (, t = e λ e t ɛτ Ũ(, t, where < λ <, such that Ũ(, t = e λ e t ɛτ Uc2 (, t, (

54 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN then (2.34 can be rewritten in terms of Ũ(, t as follows [ Ũ(, t t + 2ɛ 2 τ e +ξ ɛ τ e ξ e λ λξ ɛ τ τ Ũ(ξ, s dξ ds t + + e +ξ ɛ τ e ξ c (s e λ ξ ɛ s τ e ɛτ dξ ds t + ] + e +ξ ɛ τ e ξ e λ u (ξ dξ ds + D [ t + e +ξ 2ɛ 2 ɛ τ + sgn( ξe ξ e λ λξ ɛ τ Ũ(ξ, s dξ ds τ t + + e +ξ ɛ τ + sgn( ξe ξ c (s e λ ξ ɛ s τ e ɛτ dξ ds t + ] + e +ξ ɛ τ + sgn( ξe ξ e λ u (ξ dξ ds. Because of Lemmas , we can get the following estimate for on (2.36 : (2.36 Ũ(, t based Ũ(, t 2ɛ 2 τ τ + D 2ɛ 2 τ t [ 2ɛ τ λ 2 t +2C u e λl Ũ(, s ds + t ] ds [ 2ɛ τ t λ Ũ(, s 2 ds + +2C u ɛ t ] τe λl ds b t τ ɛτ Ũ(, s ds + ã τ (s ɛτ ɛ τ e( λ ( + t c (s e s ɛτ ds t c (s e s ɛτ ds e( λ ds, (

55 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN where b τ = + D τ λ 2, ã τ (t = a τ e t ɛτ + cτ e λl, a τ = c ( ( + D τ(e( λ +, 2e( λ c τ = C u ( + D τ. By Gronwall s inequality, inequality (2.37 gives that Ũ(, t t ã τ (t s bτ (t s e ɛτ ds ɛτ ( a τ e t t ɛτ + cτ ɛτ e λl e bτ t ɛτ. Hence U c2 (, t Ũ(, t e λ e t ɛτ ( a τ e t t ɛτ + cτ ɛτ e λl e bτ t ɛτ e λ e t ɛτ, i.e., U c2 (, t decays eponentially with respect to. In particular, when = L, we have as given in (2.32. U c2 (L, t a τ e bτ t ɛτ e λl + t cτ e (bτ t ɛτ e λ(l L (2.38 ɛτ Proof of Theorem 2.. In this section, we will first find the maimum difference of u L (, t v L (, t in proposition 2.3.5, then we will derive u L (, t v L (, t H, and W (, t L,ɛ,τ H = L,ɛ,τ 4

56 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN U(, t V (, t H L,ɛ,τ in propositions and respectively, and in turn we will get u(, t v(, t H L,ɛ,τ in theorem Proposition If u ( satisfies (2.3, then u L v L E ;ɛ,τ (te λl + E 2;ɛ,τ (te λ(l L where E ;ɛ,τ (t = c ( + a τ e bτ t ɛτ and E 2;ɛ,τ (t = c τ t ɛτ e (bτ t ɛτ. Proof. By the definition of u L and v L given in (2.22 and (2.25 and the assumption that u ( = v ( for [, L], we can get their difference u L (, t v L (, t = c (t (e ( φ ( + U c2 (L, t h(t + e t ɛτ h( φ 2 ( Combining Lemmas 2.3.4(i, 2.3.4(ii, inequality (2.38, and h(t, we have u L (, t v L (, t c (t e L + Uc2 (L, t E ;ɛ,τ (te λl + E2;ɛ,τ (te λ(l L (2.39 where E ;ɛ,τ (t = c ( + a τ e bτ t ɛτ, E 2;ɛ,τ (t = c τ t ɛτ e (bτ t ɛτ. Proposition If u ( satisfies (2.3, and E ;ɛ,τ (t, E 2;ɛ,τ (t are as in proposition 2.3.5, then u L (, t v L (, t H 5L (E ;ɛ,τ (te λl + E2;ɛ,τ (te λ(l L. L,ɛ,τ 4

57 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Proof. Because of the definition of u L and v L given in (2.22 and (2.25, Lemmas 2.3.4(iii and inequality (2.38, we have that (u L (, t v L (, t c (t e L φ 2 ( + U c2 (L, t φ 2( 2 Now, combining (2.39 and (2.4, we obtain that L (E ;ɛ,τ (te λl + E2;ɛ,τ (te λ(l L. (2.4 u L (, t v L (, t H L,ɛ,τ = 5L u L v L 2 + ɛ τ (ul v L 2 d (E ;ɛ,τ (te λl + E2;ɛ,τ (te λ(l L. (2.4 Proposition If u ( satisfies (2.3, then W (, t H L,ɛ,τ γ ;ɛ,τ (te λl + γ2;ɛ,τ (te λ(l L where the coefficients γ ;ɛ,τ (t and γ 2;ɛ,τ (t are derived as ( γ ;ɛ,τ (t = e (M+ 2 t 2M (M + 2 τ 2M ( γ 2;ɛ,τ (t = e (M+ 2 t 2M (M + 2 τ 2M ( L t + ( Lcτ + ɛτ c ( + a τ (e bτ t ɛτ, b τ t ɛτ(b τ e (bτ t ɛτ (b τ 2 (e (bτ t ɛτ. Proof. Multiplying the governing equation of W (2.28 by 2W, integrating over [, L], L = t 2W W t d ɛ L 2W W d 2W (f(v f(u d + ɛτ 42 L L ɛ 2 τ2w W t d 2W (v L u L d,

58 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN and using integration by parts, we get d dt = ɛ L L W 2 + (W 2 d 2W 2 d + L 2W (f(v f(u d + 2 ɛτ L Therefore, using the norm we defined earlier in (2.3, we have that 2 and notice that d dt W (, t 2 HL,ɛ,τ L W f (η v u d + 2 L ɛτ W (v L u L d. v L u L W (, t H L,ɛ,τ f (u (M + 2 2M we denote this upper bound by C, i.e., C = (M + 2 2M, then we have that d dt W (, t 2 HL,ɛ,τ 2C 2C L W ( W + v L u L d + 2 L v L u L ɛτ W (, t H L,ɛ,τ ( W (, t 2 H + v L,ɛ,τ L u L L W (, t H L,ɛ,τ + 2 L ɛτ = 2C W (, t 2 H + L,ɛ,τ v L u L W (, t H L,ɛ,τ ( 2C + 2 ɛτ L vl u L W (, t H. L,ɛ,τ 43

59 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Hence, d dt W (, t H L,ɛ,τ C ( C W (, t H + L,ɛ,τ + L vl u L ɛτ. By Gronwall s inequality and (2.39 W (, t H L,ɛ,τ Hence t ( C + L vl u L ɛτ e C(t s ds ( e Ct C + L t E ;ɛ,τ (se λl + E2;ɛ,τ (se λ(l L ds ɛτ ( ( e Ct C + L t E ;ɛ,τ (s ds e λl ɛτ ( ( + e Ct C + L t E 2;ɛ,τ (s ds e λ(l L ɛτ ( e Ct C + ( L t c ( + a τɛτ (e bτ t ɛτ e λl ɛτ b ( τ +e Ct C + L c τ ɛτ ɛτ ( ɛτ b τ te (bτ t ɛτ ɛτ ( b τ 2 (e (bτ t ɛτ e λ(l L. W (, t H L,ɛ,τ γ ;ɛ,τ (te λl + γ2;ɛ,τ (te λ(l L, where ( ( γ ;ɛ,τ (t = e Ct t C τ + L ɛτ c ( + a τ (e bτ t ɛτ b ( τ ( C τ + Lcτ γ 2;ɛ,τ (t = e Ct t ɛτ(b τ e (bτ t ɛτ, (b τ 2 (e (bτ t ɛτ. Now comes the main theorem of this section. 44

60 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN Theorem If u ( satisfies (2.3, and E ;ɛ,τ (t, E 2;ɛ,τ (t, γ ;ɛ,τ (t, γ 2;ɛ,τ (t are as in proposition and 2.3.7, then u(, t v(, t H D ;ɛ,τ (te λl + L,ɛ,τ D 2;ɛ,τ (te λ(l L where D ;ɛ,τ (t = γ ;ɛ,τ (t + 5LE ;ɛ,τ (t and D 2;ɛ,τ (t = γ 2;ɛ,τ (t + 5LE2;ɛ,τ (t. Proof of the Main Theorem. u(, t v(, t H L,ɛ,τ W (, t H L,ɛ,τ + v L (, t u L (, t H L,ɛ,τ = D ;ɛ,τ (te λl + D2;ɛ,τ (te λ(l L, where D ;ɛ,τ (t = γ ;ɛ,τ (t + 5LE ;ɛ,τ (t ( ( = e Ct t C τ + L ɛτ c ( + a τ (e bτ t ɛτ b τ + 5L( c( + a τ e bτ t ɛτ, D 2;ɛ,τ (t = γ 2;ɛ,τ (t + 5LE 2;ɛ,τ (t ( ( = e Ct t C τ + Lcτ ɛτ(b τ e (bτ t ɛτ + 5Lc τ t ɛτ e (bτ t ɛτ. (b τ (e (bτ t 2 ɛτ λl This theorem shows that if ɛ and λ(l L τ converge to infinity, then the solution v(, t of the finite interval boundary value problem converges to the solution u(, t of the half line problem in the sense of H L,ɛ,τ. This can be achieved either by letting L or ɛ. For eample, in the etreme case, ɛ =, the half line problem (2.2 becomes hyperbolic and the domain of dependence is finite, so, certainly, one 45

61 CHAPTER 2. HALF LINE V.S. FINITE INTERVAL DOMAIN only need to consider the finite interval boundary value problem. This is consistent with the main theorem in the sense that for a fied final time t, if λl > b τ t and λ(l L > (b τ t, i.e., L > ma( bτ t (bτ t,, then u(, t v(, t λ λ H L,ɛ,τ D ;ɛ,τ (te λl + D2;ɛ,τ (te λ(l L as ɛ. Theorem gives a theoretical justification for using the solution of the finite interval boundary value problem to approimate the solution of the half line problem with appropriate choice of L and ɛ. Hence in the net chapter, the numerical scheme designed to solve the MBL equation (.9 is given for finite interval boundary value problem. 46