Linearized Gas Dynamics Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 273 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 274 Find the Eigenvalues: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 275 Arrange the Eigenvalues from smallest to largest: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 276 Eigenvectors: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 277 Eigenvector corresponding to first Eigenvalue: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 278 Similarly for Eigenvector corresponding to second Eigenvalue: Eigenvectors: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 279 Hyerbolicity A system is hyperbolic if matrix A (m x m) has m real eigenvalues and a corresponding set of m linearly independent right eigenvectors. If the eigenvalues are all distinct, the system is strictly hyperbolic. If none of the eigenvalues are real, the system is elliptic. Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 280 Small Perturbation Equations where: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 281 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 282 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 283 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 284 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 285 Linear Advection Equation An Initial Value Problem: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 286 Characteristic Curves: Curves in the t-x plane along which a PDE becomes an ODE Regard x as a function of t: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 287 What is the rate of change of x along t? Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 288 If the characteristic curve satisfies Then Conclusion: u is constant along the characteristic curve! Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 289 Defines a 1-parameter family of curves! Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 290 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 291 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 292 The Riemann Problem Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 293 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 294 Based on our previous experience with the linear advection equation, what kind of solution do we expect? Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 295 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 296 Linear Hyperbolic Systems Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 297 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 298 Diagonalisability Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 299 Characteristic Variables Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 300 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 301 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 302 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 303 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 304 General Initial Value Problems Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 305 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 306 Linear Gas Dynamics Revisited Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 307 Initial Conditions: Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 308 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 309 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 310 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 311 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 312 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 313 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 314 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 315 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 316 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 317 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 318 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 319 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 320 Image Credits Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 321 Reading • Kreyszig, Chapter 7, 8 • Greenberg, Chapters 10, 11 Fall 2021 Applied Mathematics Lecture 02: Linear Algebra 322
