CME306 / CS205B Homework 1 (Theory)Conservation of Mass (Eulerian Framework)1. In an Eulerian framework, the strong form of Conservation of Mass takes the form below. Please brieflyexplain the three nonzer o terms in the equation.ρt+ ρux+ uρx= 0 (1)2. If we are working with a discontinuous density field or velocity field (i.e. either ρxor uxdon’t existsomewhere in the domain), we cannot use the stro ng form of Conservation of Mass . We can howeverapply the weak form, which describes how mass changes in a control volume Ω (here, mass is given asRΩρdV ). Please derive the weak form eq uation for Co ns e rvation of Mass from the strong form (notethat the weak for m should have no spatial derivatives).13. In an Euleria n framework, the graphs of ρ and u in the plots below describe the s tate of a system.Does ρ increase, decrease, or stay the same at the sample point y0for each system? (You may assumeall quantities here are positive)ρuy0(a) System 1ρuy0(b) System 2ρuy0(c) System 3ρuy0(d) System 42Convergence AnalysisConsider the wave equationut+ aux= 0where a = constant. Establish whether or not the following methods for solving the equation converge. Ifso, what are the conditions for convergence? Hint: Use the Lax-Richtmyer equivalence theorem. Chapters1 and 2 of the text by Strikwerda will be helpful, in addition to the discussion notes provided online.Note that (D+φ)i=φi+1−φi∆x, and (D−φ)i=φi−φi−1∆x.1. Explicit Central Differencingvn+1j− vnj∆t+ avnj+1− vnj−12∆x= 0.2. Implicit Central Differencingvn+1j− vnj∆t+ avn+1j+1− vn+1j−12∆x= 0.3. Upwindingvn+1j− vnj∆t+ aD∗vnj= 0If a > 0, D∗= D−. If a < 0, D∗= D+.4. Downwindingvn+1j− vnj∆t+ aD∗vnj= 0If a > 0, D∗= D+. If a < 0, D∗=
View Full Document
Unlocking...