Quasi-linear first order equations. Consider the nonlinear transport equation
|
|
|
- Erica Evans
- 5 years ago
- Views:
Transcription
1 Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x <
2 Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x < If we (somehow) knew u(x, t), could find characteristics by solving X (T ) = c(u(x(t ), T )), X(t) = x.
3 Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x < If we (somehow) knew u(x, t), could find characteristics by solving X (T ) = c(u(x(t ), T )), X(t) = x. Note on each characteristic, solution u(x, t) = U(T ) = u(x(0), 0) = f (X(0)) is just constant.
4 Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x < If we (somehow) knew u(x, t), could find characteristics by solving X (T ) = c(u(x(t ), T )), X(t) = x. Note on each characteristic, solution u(x, t) = U(T ) = u(x(0), 0) = f (X(0)) is just constant. Since u is not really known in advance, what is initial position of the characteristic X(0)?
5 Quasi-linear first order equations Consider the nonlinear transport equation u t + c(u)u x = 0, u(x, 0) = f (x) < x < If we (somehow) knew u(x, t), could find characteristics by solving X (T ) = c(u(x(t ), T )), X(t) = x. Note on each characteristic, solution u(x, t) = U(T ) = u(x(0), 0) = f (X(0)) is just constant. Since u is not really known in advance, what is initial position of the characteristic X(0)? Idea: find characteristic X(T ) and the solution on characteristic U simultaneously, in a way that is compatible with the initial condition.
6 Example Solve u t + 3uu x = 0, u(x, 0) = x.
7 Example Solve u t + 3uu x = 0, u(x, 0) = x. Characteristics solve X (T ) = 3U = constant subject to X(t) = x.
8 Example Solve u t + 3uu x = 0, u(x, 0) = x. Characteristics solve X (T ) = 3U = constant subject to X(t) = x. Therefore X(T ) = 3U(T t) + x.
9 Example Solve u t + 3uu x = 0, u(x, 0) = x. Characteristics solve X (T ) = 3U = constant subject to X(t) = x. Therefore X(T ) = 3U(T t) + x. At initial position X(0) = x 3Ut, U must agree with initial condition U = u(x(0), 0) = X(0) = x 3Ut.
10 Example Solve u t + 3uu x = 0, u(x, 0) = x. Characteristics solve X (T ) = 3U = constant subject to X(t) = x. Therefore X(T ) = 3U(T t) + x. At initial position X(0) = x 3Ut, U must agree with initial condition U = u(x(0), 0) = X(0) = x 3Ut. Solving for U gives U = u(x, t) = x 1 + 3t.
11 Quasi-linear first order equations Notice for equations of the form u t + c(u)u x = 0, characteristics are always straight lines.
12 Quasi-linear first order equations Notice for equations of the form u t + c(u)u x = 0, characteristics are always straight lines. Two possible problems: 1 Characteristics can intersect. This produces shock waves.
13 Quasi-linear first order equations Notice for equations of the form u t + c(u)u x = 0, characteristics are always straight lines. Two possible problems: 1 Characteristics can intersect. This produces shock waves. 2 Characteristics don t necessarily pass through every point (x, t) for t > 0. This produces rarefaction waves.
14 Shocks Consider u t + uu x = 0, u(x, 0) = { 2 x < 0 1 x > 0 For x < 0 have speed = 2, for x > 0 speed = 1, and characteristics intersect in region t < x < 2t.
15 Shocks Consider u t + uu x = 0, u(x, 0) = { 2 x < 0 1 x > 0 For x < 0 have speed = 2, for x > 0 speed = 1, and characteristics intersect in region t < x < 2t. Resolution: allow the solution to be discontinuous: { 2 x < x s (t) u(x, t) = 1 x > x s (t) where x s (t) is a curve called a shock. But what determines x s?
16 Shock speed Let the shock speed be x s(t) = c s, and rewrite equation in conservation form u t + J(u) x = 0, J (u) = c(u).
17 Shock speed Let the shock speed be x s(t) = c s, and rewrite equation in conservation form u t + J(u) x = 0, J (u) = c(u). Look for solutions u = U(x c s t) which are waves traveling at speed c s, with { u L x U x +. u R
18 Shock speed Let the shock speed be x s(t) = c s, and rewrite equation in conservation form u t + J(u) x = 0, J (u) = c(u). Look for solutions u = U(x c s t) which are waves traveling at speed c s, with { u L x U x +. Plugging into u t + J(u) x = 0, u R c s U + J(U) = 0. Integration over R gives c s (u R u L ) + J(u R ) J(u L ) = 0 or c s = J(u R) J(u L ) u R u L, (Rankine-Hugoniot condition)
19 Example, continued For u t + uu x = 0, u(x, 0) = { 2 x < 0 1 x > 0 we have u(x, t) = { 2 x < x s (t) 1 x > x s (t).
20 Example, continued For u t + uu x = 0, u(x, 0) = { 2 x < 0 1 x > 0 we have u(x, t) = { 2 x < x s (t) 1 x > x s (t). Equation in conservation form is u t + (u 2 /2) x = 0, therefore J(u) = u 2 /2.
21 Example, continued For u t + uu x = 0, u(x, 0) = { 2 x < 0 1 x > 0 we have u(x, t) = { 2 x < x s (t) 1 x > x s (t). Equation in conservation form is u t + (u 2 /2) x = 0, therefore J(u) = u 2 /2. Shock evolves according to dx s /dt = [J(2) J(1)]/(2 1) = 3/2, x s (0) = 0, so that x s (t) = 3/2t.
22 Rarefactions Now consider u t + uu x = 0, u(x, 0) = { 1 x < 0 2 x > 0 Problem: no characteristics fill region t < x < 2t, called a rarefaction.
23 Rarefactions Now consider u t + uu x = 0, u(x, 0) = { 1 x < 0 2 x > 0 Problem: no characteristics fill region t < x < 2t, called a rarefaction. Resolution: employ principle characteristics cannot emerge from other characteristics" This means that the characteristics in the rarefaction must all start at the point of discontinuity x = 0.
24 Rarefactions Now consider u t + uu x = 0, u(x, 0) = { 1 x < 0 2 x > 0 Problem: no characteristics fill region t < x < 2t, called a rarefaction. Resolution: employ principle characteristics cannot emerge from other characteristics" This means that the characteristics in the rarefaction must all start at the point of discontinuity x = 0. In the above example, rarefaction characteristics solve X (T ) = U = constant, subject to both X(0) = 0 and X(t) = x.
25 Rarefactions Now consider u t + uu x = 0, u(x, 0) = { 1 x < 0 2 x > 0 Problem: no characteristics fill region t < x < 2t, called a rarefaction. Resolution: employ principle characteristics cannot emerge from other characteristics" This means that the characteristics in the rarefaction must all start at the point of discontinuity x = 0. In the above example, rarefaction characteristics solve X (T ) = U = constant, subject to both X(0) = 0 and X(t) = x. Solution: X(T ) = UT, which at T = t gives x = Ut or U = x/t.
26 Rarefactions Now consider u t + uu x = 0, u(x, 0) = { 1 x < 0 2 x > 0 Problem: no characteristics fill region t < x < 2t, called a rarefaction. Resolution: employ principle characteristics cannot emerge from other characteristics" This means that the characteristics in the rarefaction must all start at the point of discontinuity x = 0. In the above example, rarefaction characteristics solve X (T ) = U = constant, subject to both X(0) = 0 and X(t) = x. Solution: X(T ) = UT, which at T = t gives x = Ut or U = x/t. Therefore complete solution is 1 x < t u(x, t) = x/t t < x < 2t 2 x > 2t.
27 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1.
28 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1. Three regions: x < 0, 0 < x < x s (t), and x > x s (t).
29 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1. Three regions: x < 0, 0 < x < x s (t), and x > x s (t). Solution is only non-zero in second region. Characteristics there solve X (T ) = U subject to X(t) = x, which has a solution X(T ) = U(T t) + x, so that X(0) = x Ut.
30 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1. Three regions: x < 0, 0 < x < x s (t), and x > x s (t). Solution is only non-zero in second region. Characteristics there solve X (T ) = U subject to X(t) = x, which has a solution X(T ) = U(T t) + x, so that X(0) = x Ut. Since U = u(x(0), 0) = 2X(0), it follows that 2(x Ut) = U or u(x, t) = U = 2x/(1 + 2t).
31 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1. Three regions: x < 0, 0 < x < x s (t), and x > x s (t). Solution is only non-zero in second region. Characteristics there solve X (T ) = U subject to X(t) = x, which has a solution X(T ) = U(T t) + x, so that X(0) = x Ut. Since U = u(x(0), 0) = 2X(0), it follows that 2(x Ut) = U or u(x, t) = U = 2x/(1 + 2t). Conservation form is u t + J(u) x = 0 with J = u 2 /2, so x s(t) = J(2x s/(1 + 2t)) J(0) 2x s /(1 + 2t) 0 = x s 1 + 2t.
32 Example 1 (non-constant shock speed). Consider 0 x < 0 u t + uu x = 0, u(x, 0) = 2x 0 < x < 1 0 x > 1. Three regions: x < 0, 0 < x < x s (t), and x > x s (t). Solution is only non-zero in second region. Characteristics there solve X (T ) = U subject to X(t) = x, which has a solution X(T ) = U(T t) + x, so that X(0) = x Ut. Since U = u(x(0), 0) = 2X(0), it follows that 2(x Ut) = U or u(x, t) = U = 2x/(1 + 2t). Conservation form is u t + J(u) x = 0 with J = u 2 /2, so x s(t) = J(2x s/(1 + 2t)) J(0) 2x s /(1 + 2t) 0 = x s 1 + 2t. Solving this differential equation with initial condition x s (0) = 1 gives x s (t) = 1 + 2t.
33 Example 2 (shock creation) Consider 1 x < 0 u t + (u + 1)u x = 0, u(x, 0) = 1 x 0 < x < 1 0 x > 1. Characteristics which start at x = 0 and x = 1 are lines X 1 (T ) = 2T and X 2 (T ) = 1 + T.
34 Example 2 (shock creation) Consider 1 x < 0 u t + (u + 1)u x = 0, u(x, 0) = 1 x 0 < x < 1 0 x > 1. Characteristics which start at x = 0 and x = 1 are lines X 1 (T ) = 2T and X 2 (T ) = 1 + T. These intersect at X = 2 and T = 1 to create shock, evolving as x s(t) = so that x s (t) = 3/2(t 1) + 2. J(0) J(1), J(u) = u 2 /2 + u, x c (1) =
35 Example 2 (shock creation) Consider 1 x < 0 u t + (u + 1)u x = 0, u(x, 0) = 1 x 0 < x < 1 0 x > 1. Characteristics which start at x = 0 and x = 1 are lines X 1 (T ) = 2T and X 2 (T ) = 1 + T. These intersect at X = 2 and T = 1 to create shock, evolving as x s(t) = so that x s (t) = 3/2(t 1) + 2. J(0) J(1), J(u) = u 2 /2 + u, x c (1) = In region X 1 (t) < x < X 2 (t), characteristics with X(t) = x satisfy X (T ) = (U + 1).
36 Example 2 (shock creation) Consider 1 x < 0 u t + (u + 1)u x = 0, u(x, 0) = 1 x 0 < x < 1 0 x > 1. Characteristics which start at x = 0 and x = 1 are lines X 1 (T ) = 2T and X 2 (T ) = 1 + T. These intersect at X = 2 and T = 1 to create shock, evolving as x s(t) = so that x s (t) = 3/2(t 1) + 2. J(0) J(1), J(u) = u 2 /2 + u, x c (1) = In region X 1 (t) < x < X 2 (t), characteristics with X(t) = x satisfy X (T ) = (U + 1). They are therefore lines X(T ) = (U + 1)(T t) + x.
37 Example 2 (shock creation) Consider 1 x < 0 u t + (u + 1)u x = 0, u(x, 0) = 1 x 0 < x < 1 0 x > 1. Characteristics which start at x = 0 and x = 1 are lines X 1 (T ) = 2T and X 2 (T ) = 1 + T. These intersect at X = 2 and T = 1 to create shock, evolving as x s(t) = so that x s (t) = 3/2(t 1) + 2. J(0) J(1), J(u) = u 2 /2 + u, x c (1) = In region X 1 (t) < x < X 2 (t), characteristics with X(t) = x satisfy X (T ) = (U + 1). They are therefore lines X(T ) = (U + 1)(T t) + x. To be consistent with the initial condition, U = u(x(0), 0) = 1 [x (U + 1)t], so that U = u(x, t) = 1 x + t. 1 t Notice that this solution breaks down at t = 1.
38 Example 3 (multiple shocks) Consider 3 x < 1 u t + 3u 2 u x = 0, u(x, 0) = 2 1 < x < 2 1 x > 2. For small t, three regions with characteristic speeds 27 x < x 1 (t) X (T ) = 3U 2 = 12 x 1 (t) < x < x 2 (t) 3 x > x 2 (t).
39 Example 3 (multiple shocks) Consider 3 x < 1 u t + 3u 2 u x = 0, u(x, 0) = 2 1 < x < 2 1 x > 2. For small t, three regions with characteristic speeds 27 x < x 1 (t) X (T ) = 3U 2 = 12 x 1 (t) < x < x 2 (t) 3 x > x 2 (t). Conservation form u t + J(u) x = 0 has J(u) = u 3, so that x 1 (t) = = 19, x 1(0) = 1 x 2 (t) = = 7, x 2(0) = 2
40 Example 4,cont. Solution: x 1 (t) = 19t + 1, x 2 (t) = 7t + 1.
41 Example 4,cont. Solution: x 1 (t) = 19t + 1, x 2 (t) = 7t + 1. These intersect where t = 1/12 and x = 31/12, producing a single shock with u = 3 on left and u = 1 on right.
42 Example 4,cont. Solution: x 1 (t) = 19t + 1, x 2 (t) = 7t + 1. These intersect where t = 1/12 and x = 31/12, producing a single shock with u = 3 on left and u = 1 on right. Third shock x 3 (t) evolves according to x 3 (t) = = 13, x 3(1/12) = 32/12, so that x 3 (t) = 13(t 1/12) + 31/12.
Modeling Rarefaction and Shock waves
April 30, 2013 Inroduction Inroduction Rarefaction and shock waves are combinations of two wave fronts created from the initial disturbance of the medium. Inroduction Rarefaction and shock waves are combinations
A Very Brief Introduction to Conservation Laws
A Very Brief Introduction to Wen Shen Department of Mathematics, Penn State University Summer REU Tutorial, May 2013 Summer REU Tutorial, May 2013 1 / The derivation of conservation laws A conservation
Notes: Outline. Shock formation. Notes: Notes: Shocks in traffic flow
Outline Scalar nonlinear conservation laws Traffic flow Shocks and rarefaction waves Burgers equation Rankine-Hugoniot conditions Importance of conservation form Weak solutions Reading: Chapter, 2 R.J.
0.3.4 Burgers Equation and Nonlinear Wave
16 CONTENTS Solution to step (discontinuity) initial condition u(x, 0) = ul if X < 0 u r if X > 0, (80) u(x, t) = u L + (u L u R ) ( 1 1 π X 4νt e Y 2 dy ) (81) 0.3.4 Burgers Equation and Nonlinear Wave
Math Partial Differential Equations 1
Math 9 - Partial Differential Equations Homework 5 and Answers. The one-dimensional shallow water equations are h t + (hv) x, v t + ( v + h) x, or equivalently for classical solutions, h t + (hv) x, (hv)
Waves in a Shock Tube
Waves in a Shock Tube Ivan Christov c February 5, 005 Abstract. This paper discusses linear-wave solutions and simple-wave solutions to the Navier Stokes equations for an inviscid and compressible fluid
Numerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear
Lecture 5.7 Compressible Euler Equations
Lecture 5.7 Compressible Euler Equations Nomenclature Density u, v, w Velocity components p E t H u, v, w e S=c v ln p - c M Pressure Total energy/unit volume Total enthalpy Conserved variables Internal
IV. Compressible flow of inviscid fluids
IV. Compressible flow of inviscid fluids Governing equations for n = 0, r const: + (u )=0 t u + ( u ) u= p t De e = + ( u ) e= p u+ ( k T ) Dt t p= p(, T ), e=e (,T ) Alternate forms of energy equation
Non-linear Scalar Equations
Non-linear Scalar Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro August 24, 2014 1 / 44 Overview Here
Shock Waves. 1 Steepening of sound waves. We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: kˆ.
Shock Waves Steepening of sound waves We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: v u kˆ c s kˆ where u is the velocity of the fluid and k is the wave
Shock on the left: locus where cars break behind the light.
Review/recap of theory so far. Evolution of wave profile, as given by the characteristic solution. Graphical interpretation: Move each point on graph at velocity c(ρ). Evolution as sliding of horizontal
MATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
Answers to Problem Set Number 04 for MIT (Spring 2008)
Answers to Problem Set Number 04 for 18.311 MIT (Spring 008) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139). March 17, 008. Course TA: Timothy Nguyen, MIT, Dept. of Mathematics, Cambridge,
Info. No lecture on Thursday in a week (March 17) PSet back tonight
Lecture 0 8.086 Info No lecture on Thursday in a week (March 7) PSet back tonight Nonlinear transport & conservation laws What if transport becomes nonlinear? Remember: Nonlinear transport A first attempt
Salmon: Lectures on partial differential equations
4 Burger s equation In Lecture 2 we remarked that if the coefficients in u x, y,! "! "x + v x,y,! "! "y = 0 depend not only on x,y but also on!, then the characteristics may cross and the solutions become
Modeling using conservation laws. Let u(x, t) = density (heat, momentum, probability,...) so that. u dx = amount in region R Ω. R
Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...) so that u dx = amount in region R Ω. R Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...)
Numerical Methods for Hyperbolic Conservation Laws Lecture 4
Numerical Methods for Hyperbolic Conservation Laws Lecture 4 Wen Shen Department of Mathematics, Penn State University Email: wxs7@psu.edu Oxford, Spring, 018 Lecture Notes online: http://personal.psu.edu/wxs7/notesnumcons/
Coupling conditions for transport problems on networks governed by conservation laws
Coupling conditions for transport problems on networks governed by conservation laws Michael Herty IPAM, LA, April 2009 (RWTH 2009) Transport Eq s on Networks 1 / 41 Outline of the Talk Scope: Boundary
Notes for Introduction to Partial Di erential Equations, Fall 2018
Notes for Introduction to Partial Di erential Equations, Fall 2018 Esteban G. Tabak 1 Tra c Flow We eplore a simple mathematical model for tra c flow. This will be our introduction to systems of conservation
The second-order 1D wave equation
C The second-order D wave equation C. Homogeneous wave equation with constant speed The simplest form of the second-order wave equation is given by: x 2 = Like the first-order wave equation, it responds
AMath 574 February 11, 2011
AMath 574 February 11, 2011 Today: Entropy conditions and functions Lax-Wendroff theorem Wednesday February 23: Nonlinear systems Reading: Chapter 13 R.J. LeVeque, University of Washington AMath 574, February
MATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, < 1, 1 +, 1 < < 0, ψ() = 1, 0 < < 1, 0, > 1, so that it verifies ψ 0, ψ() = 0 if 1 and ψ()d = 1. Consider (ψ j ) j 1 constructed as
Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3
Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Tommaso Ruggeri Department of Mathematics and Research Center of Applied Mathematics University of Bologna January 21, 2017 ommaso
Finite difference method for solving Advection-Diffusion Problem in 1D
Finite difference method for solving Advection-Diffusion Problem in 1D Author : Osei K. Tweneboah MATH 5370: Final Project Outline 1 Advection-Diffusion Problem Stationary Advection-Diffusion Problem in
The first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
A review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler:
Lecture 7 18.086 Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler: U j,n+1 t U j,n = U j+1,n 2U j,n + U j 1,n x 2 Expected accuracy: O(Δt) in time,
2-Motivations From the Theory of ODE s MATH 22C
2-Motivations From the Theory of ODE s MATH 22C 1. Ordinary Differential Equations (ODE) and the Fundamental Role of the Derivative in the Sciences Recall that a real valued function of a real variable
Conservation and dissipation principles for PDEs
Conservation and dissipation principles for PDEs Modeling through conservation laws The notion of conservation - of number, energy, mass, momentum - is a fundamental principle that can be used to derive
Various lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
arxiv: v2 [math.ap] 1 Jul 2011
![arxiv: v2 [math.ap] 1 Jul 2011](/thumbs/92/110503308.jpg "arxiv: v2 [math.ap] 1 Jul 2011")
A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime arxiv:1105.3074v2 [math.ap] 1 Jul 2011 Abstract Philippe G. efloch 1 and Mai Duc Thanh 2 1 aboratoire
Preparation for the Final
Preparation for the Final Basic Set of Problems that you should be able to do: - all problems on your tests (- 3 and their samples) - ex tra practice problems in this documents. The final will be a mix
Nonlinear stability of compressible vortex sheets in two space dimensions
of compressible vortex sheets in two space dimensions J.-F. Coulombel (Lille) P. Secchi (Brescia) CNRS, and Team SIMPAF of INRIA Futurs Evolution Equations 2006, Mons, August 29th Plan 1 2 3 Related problems
Tackling the Calculator Multiple Choice Section (AB)
Tackling the Calculator Multiple Choice Section (AB) The approach to attacking the calculator multiple choice problems on the AP exam can be very different than the non-calculator multiple choice. If thought
Lecture Notes on Hyperbolic Conservation Laws
Lecture Notes on Hyperbolic Conservation Laws Alberto Bressan Department of Mathematics, Penn State University, University Park, Pa. 16802, USA. bressan@math.psu.edu May 21, 2009 Abstract These notes provide
Hyperbolic Systems of Conservation Laws. in One Space Dimension. I - Basic concepts. Alberto Bressan. Department of Mathematics, Penn State University
Hyperbolic Systems of Conservation Laws in One Space Dimension I - Basic concepts Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1 The Scalar Conservation
Finite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
Math 660-Lecture 23: Gudonov s method and some theories for FVM schemes
Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,
Rarefaction wave interaction for the unsteady transonic small disturbance equations
Rarefaction wave interaction for the unsteady transonic small disturbance equations Jun Chen University of Houston Department of Mathematics 4800 Calhoun Road Houston, TX 77204, USA chenjun@math.uh.edu
Math 124A October 11, 2011
Math 14A October 11, 11 Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This corresponds to a string of infinite length. Although
The Discontinuous Galerkin Method for Hyperbolic Problems
Chapter 2 The Discontinuous Galerkin Method for Hyperbolic Problems In this chapter we shall specify the types of problems we consider, introduce most of our notation, and recall some theory on the DG
Chapter 2: Motion along a straight line
Chapter 2: Motion along a straight line This chapter uses the definitions of length and time to study the motions of particles in space. This task is at the core of physics and applies to all objects irregardless
1 Basic Second-Order PDEs
Partial Differential Equations A. Visintin a.a. 2011-12 These pages are in progress. They contain: an abstract of the classes; notes on some (few) specific issues. These notes are far from providing a
Updated: January 16, 2016 Calculus II 7.4. Math 230. Calculus II. Brian Veitch Fall 2015 Northern Illinois University
Math 30 Calculus II Brian Veitch Fall 015 Northern Illinois University Integration of Rational Functions by Partial Fractions From algebra, we learned how to find common denominators so we can do something
( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
Vector Spaces. EXAMPLE: Let R n be the set of all n 1 matrices. x 1 x 2. x n
Vector Spaces DEFINITION: A vector space is a nonempty set V of ojects, called vectors, on which are defined two operations, called addition and multiplication y scalars (real numers), suject to the following
ON SCALAR CONSERVATION LAWS WITH POINT SOURCE AND STEFAN DIEHL
ON SCALAR CONSERVATION LAWS WITH POINT SOURCE AND DISCONTINUOUS FLUX FUNCTION STEFAN DIEHL Abstract. The conservation law studied is @u(x;t) + @ F u(x; t); x @t @x = s(t)(x), where u is a f(u); x > 0 concentration,
y + p(t)y = g(t) for each t I, and that also satisfies the initial condition y(t 0 ) = y 0 where y 0 is an arbitrary prescribed initial value.
p1 Differences Between Linear and Nonlinear Equation Theorem 1: If the function p and g are continuous on an open interval I : α < t < β containing the point t = t 0, then there exists a unique function
Section Vector Functions and Space Curves
Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of
The Riemann problem. The Riemann problem Rarefaction waves and shock waves
The Riemann problem Rarefaction waves and shock waves 1. An illuminating example A Heaviside function as initial datum Solving the Riemann problem for the Hopf equation consists in describing the solutions
ENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
Lecture 10: Whitham Modulation Theory
Lecture 10: Whitham Modulation Theory Lecturer: Roger Grimshaw. Write-up: Andong He June 19, 2009 1 Introduction The Whitham modulation theory provides an asymptotic method for studying slowly varying
Scalar Conservation Laws and First Order Equations Introduction. Consider equations of the form. (1) u t + q(u) x =0, x R, t > 0.
Scalar Conservation Laws and First Order Equations Introduction. Consider equations of the form (1) u t + q(u) x =, x R, t >. In general, u = u(x, t) represents the density or the concentration of a physical
I N T R O D U C T I O N T O P A R T I A L D I F F E R E N T I A L E Q U AT I O N S
R U S S E L L L. H E R M A N I N T R O D U C T I O N T O P A R T I A L D I F F E R E N T I A L E Q U AT I O N S R. L. H E R M A N - V E R S I O N D AT E : S E P T E M B E R 3, 4 Copyright 4 by Russell
Lecture Notes on Numerical Schemes for Flow and Transport Problems
Lecture Notes on Numerical Schemes for Flow and Transport Problems by Sri Redeki Pudaprasetya sr pudap@math.itb.ac.id Department of Mathematics Faculty of Mathematics and Natural Sciences Bandung Institute
An Introduction to Numerical Methods for Differential Equations. Janet Peterson
An Introduction to Numerical Methods for Differential Equations Janet Peterson Fall 2015 2 Chapter 1 Introduction Differential equations arise in many disciplines such as engineering, mathematics, sciences
Semigroup Generation
Semigroup Generation Yudi Soeharyadi Analysis & Geometry Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung WIDE-Workshoop in Integral and Differensial Equations 2017
Radon measure solutions for scalar. conservation laws. A. Terracina. A.Terracina La Sapienza, Università di Roma 06/09/2017
Radon measure A.Terracina La Sapienza, Università di Roma 06/09/2017 Collaboration Michiel Bertsch Flavia Smarrazzo Alberto Tesei Introduction Consider the following Cauchy problem { ut + ϕ(u) x = 0 in
Lecture Notes on Numerical Schemes for Flow and Transport Problems
Lecture Notes on Numerical Schemes for Flow and Transport Problems by Sri Redeki Pudaprasetya sr pudap@math.itb.ac.id Department of Mathematics Faculty of Mathematics and Natural Sciences Bandung Institute
Chp 4: Non-linear Conservation Laws; the Scalar Case. By Prof. Dinshaw S. Balsara
Chp 4: Non-linear Conservation Laws; the Scalar Case By Prof. Dinshaw S. Balsara 1 4.1) Introduction We have seen that monotonicity preserving reconstruction and iemann solvers are essential building blocks
A stochastic particle system for the Burgers equation.
A stochastic particle system for the Burgers equation. Alexei Novikov Department of Mathematics Penn State University with Gautam Iyer (Carnegie Mellon) supported by NSF Burgers equation t u t + u x u
Notes: Outline. Shallow water equations. Notes: Shallow water equations. Notes:
Outline Nonlinear hyperbolic systems Shallow water equations Shock waves and Hugoniot loci Integral curves in phase plane Compression and rarefaction R.J. LeVeque, University of Washington IPDE 2011, July
EE 16B Midterm 2, March 21, Name: SID #: Discussion Section and TA: Lab Section and TA: Name of left neighbor: Name of right neighbor:
EE 16B Midterm 2, March 21, 2017 Name: SID #: Discussion Section and TA: Lab Section and TA: Name of left neighbor: Name of right neighbor: Important Instructions: Show your work. An answer without explanation
Traffic Flow I. Kurt Bryan
Introduction Traffic Flow I Kurt Bryan Consider a stretch of freeway with no entrance or eit ramps. On such a stretch of freeway we clearly have conservation of cars, barring any wrecks. Thus in our conservation
Stability of Patterns
Stability of Patterns Steve Schecter Mathematics Department North Carolina State University 1 2 Overview Motivation Systems of viscous conservation laws and their variants are partial differential equations
Hyperbolic Systems of Conservation Laws. I - Basic Concepts
Hyperbolic Systems of Conservation Laws I - Basic Concepts Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Hyperbolic Systems of Conservation Laws 1 / 27 The
Math 211. Lecture #6. Linear Equations. September 9, 2002
1 Math 211 Lecture #6 Linear Equations September 9, 2002 2 Air Resistance 2 Air Resistance Acts in the direction opposite to the velocity. 2 Air Resistance Acts in the direction opposite to the velocity.
S. N. Kruzhkov s lectures on first-order quasilinear PDEs
to appear in: Analytical and Numerical Aspects of PDEs de Gruyter 2009 S. N. Kruzhkov s lectures on first-order quasilinear PDEs Gregory A. Chechkin and Andrey Yu. Goritsky Abstract. The present contribution
CapSel Euler The Euler equations. conservation laws for 1D dynamics of compressible gas. = 0 m t + (m v + p) x
CapSel Euler - 01 The Euler equations keppens@rijnh.nl conservation laws for 1D dynamics of compressible gas ρ t + (ρ v) x = 0 m t + (m v + p) x = 0 e t + (e v + p v) x = 0 vector of conserved quantities
Scalar conservation laws with moving density constraints arising in traffic flow modeling
Scalar conservation laws with moving density constraints arising in traffic flow modeling Maria Laura Delle Monache Email: maria-laura.delle monache@inria.fr. Joint work with Paola Goatin 14th International
Introduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19 Introduction The derivation of the heat
Picard s Existence and Uniqueness Theorem
Picard s Existence and Uniqueness Theorem Denise Gutermuth 1 These notes on the proof of Picard s Theorem follow the text Fundamentals of Differential Equations and Boundary Value Problems, 3rd edition,
Hyperbolic Conservation Laws Past and Future
Hyperbolic Conservation Laws Past and Future Barbara Lee Keyfitz Fields Institute and University of Houston bkeyfitz@fields.utoronto.ca Research supported by the US Department of Energy, National Science
MATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
Solutions in the sense of distributions. Solutions in the sense of distributions Definition, non uniqueness
Solutions in the sense of distributions Definition, non uniqueness 1. Notion of distributions In order to build weak solutions to the Hopf equation, we need to define derivatives of non smooth functions,
Shock and Rarefaction Waves in a Hyperbolic Model of Incompressible Fluids
Shock and Rarefaction Waves in a Hyperbolic Model of Incompressible Fluids Andrea Mentrelli Department of Mathematics & Research Center of Applied Mathematics (CIRAM) University of Bologna, Italy Summary
Predator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline 1 Showing The PP
FDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
The Basics of Physics with Calculus Part II. AP Physics C
The Basics of Physics with Calculus Part II AP Physics C The AREA We have learned that the rate of change of displacement is defined as the VELOCITY of an object. Consider the graph below v v t lim 0 dx
Numerische Mathematik
Numer. Math. (2007 106:369 425 DOI 10.1007/s00211-007-0069-y Numerische Mathematik Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem Matania Ben-Artzi Jiequan Li Received:
Chapter 3 Burgers Equation
Chapter 3 Burgers Equation One of the major challenges in the field of comple systems is a thorough understanding of the phenomenon of turbulence. Direct numerical simulations (DNS) have substantially
1 Limits and continuity
1 Limits and continuity Question 1. Which of the following its can be evaluated by continuity ( plugging in )? sin(x) (a) x + 1 (d) x 3 x 2 + x 6 (b) e x sin(x) (e) x 2 + x 6 (c) x 2 x 2 + x 6 (f) n (
Structurally Stable Singularities for a Nonlinear Wave Equation
Structurally Stable Singularities for a Nonlinear Wave Equation Alberto Bressan, Tao Huang, and Fang Yu Department of Mathematics, Penn State University University Park, Pa. 1682, U.S.A. e-mails: bressan@math.psu.edu,
Some Aspects of Solutions of Partial Differential Equations
Some Aspects of Solutions of Partial Differential Equations K. Sakthivel Department of Mathematics Indian Institute of Space Science & Technology(IIST) Trivandrum - 695 547, Kerala Sakthivel@iist.ac.in
arxiv: v1 [math.ap] 29 May 2018
![arxiv: v1 [math.ap] 29 May 2018](/thumbs/94/121524963.jpg "arxiv: v1 [math.ap] 29 May 2018")
Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in D arxiv:805.354v [math.ap] 9 May 08 Hind Al Baba Christian Klingenberg Ondřej Kreml Václav Mácha Simon Markfelder
Propagation of discontinuities in solutions of First Order Partial Differential Equations
Propagation of discontinuities in solutions of First Order Partial Differential Equations Phoolan Prasad Department of Mathematics Indian Institute of Science, Bangalore 560 012 E-mail: prasad@math.iisc.ernet.in
On the Cauchy Problems for Polymer Flooding with Gravitation
On the Cauchy Problems for Polymer Flooding with Gravitation Wen Shen Mathematics Department, Penn State University Abstract We study two systems of conservation laws for polymer flooding in secondary
Math Homework 3 Solutions. (1 y sin x) dx + (cos x) dy = 0. = sin x =
2.6 #10: Determine if the equation is exact. If so, solve it. Math 315-01 Homework 3 Solutions (1 y sin x) dx + (cos x) dy = 0 Solution: Let P (x, y) = 1 y sin x and Q(x, y) = cos x. Note P = sin x = Q
Calculus II (Fall 2015) Practice Problems for Exam 1
Calculus II (Fall 15) Practice Problems for Exam 1 Note: Section divisions and instructions below are the same as they will be on the exam, so you will have a better idea of what to expect, though I will
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation
4.5 Linearization Calculus 4.5 LINEARIZATION. Notecards from Section 4.5: Linearization; Differentials. Linearization
4.5 Linearization Calculus 4.5 LINEARIZATION Notecards from Section 4.5: Linearization; Differentials Linearization The goal of linearization is to approximate a curve with a line. Why? Because it s easier
Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang
Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations
z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
Advection / Hyperbolic PDEs. PHY 604: Computational Methods in Physics and Astrophysics II
Advection / Hyperbolic PDEs Notes In addition to the slides and code examples, my notes on PDEs with the finite-volume method are up online: https://github.com/open-astrophysics-bookshelf/numerical_exercises
The free electron. EE 439 free electrons & scattering
The free electron In order to develop our practical understanding with quantum mechanics, we ll start with some simpler one-dimensional (function of x only), timeindependent problems. At one time, such
SHOCK WAVES FOR RADIATIVE HYPERBOLIC ELLIPTIC SYSTEMS
SHOCK WAVES FOR RADIATIVE HYPERBOLIC ELLIPTIC SYSTEMS CORRADO LATTANZIO, CORRADO MASCIA, AND DENIS SERRE Abstract. The present paper deals with the following hyperbolic elliptic coupled system, modelling
arxiv: v1 [math.ds] 17 Mar 2017
![arxiv: v1 [math.ds] 17 Mar 2017](/thumbs/95/125057018.jpg "arxiv: v1 [math.ds] 17 Mar 2017")
To appear in Dynamics of Continuous, Discrete and Impulsive Systems http:monotone.uwaterloo.ca/ journal Dwell time for switched systems with multiple equilibria on a finite time-interval Oleg Makarenkov
Optimal Control. Lecture 18. Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen. March 29, Ref: Bryson & Ho Chapter 4.
Optimal Control Lecture 18 Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen Ref: Bryson & Ho Chapter 4. March 29, 2004 Outline Hamilton-Jacobi-Bellman (HJB) Equation Iterative solution of HJB Equation
Predator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline Showing The PP Trajectories