University:
Massachusetts Institute of Technology
Course:
16.920J | Numerical Methods for Partial Differential Equations (SMA 5212)
Academic year:

2024
Views:

58

Pages:

25
Author:

Hilifrurisv1

Hyperbolic Equations: Scalar One-Dimensional Conservation Laws Lecture 11 1 Scalar Conservation Laws 1.1 Definitions 1.1.1 Conservative Form General form (ID): partial u partial t + partial f(u) partial x =0 u(x, t) is the unknown conserved quantity (mass, momentum, heat,...) (f(u)) / i * s the flux 1.1.2 Primitive Form Can also be written partial u partial t + partial f(u) partial x = partial u partial t + (df)/(du) partial u partial x =0 partial u partial t + a(u) partial u partial x =0 where a(u) = d/du (f) N1 Note 1 More General Conservation Laws In some applications, the flux function / may depend explicitly (not through u) on z; i.e. f(u, x) In such cases, the primitive form of the equation becomes partial u partial t + a(u) partial u partial x =g(u) where a(u) = partial f partial u ; and g(u)=- - plays the role of a source term. The presented here will be generally applicable, sometimes with small modifications, to this more general form. However, for clarity of presen tation we will restrict ourselves to the case where / can be determined once u is known. 1 withs s = (u_{R} + u_{L}) / 2 Note that w(y) -> u_{L} as y -> - ∞ and w(y) -> u_{R} as y -> ∞ solution and it epsilon -> 0 3.2 The Riemann Problem. In order to understand the behaviour of the solution at discontinuities it is useful to start with a simplified problem. The Riemann problem is a conservation law together with piecewise constant data having a single discontinuity. partial u partial t + partial f(u) partial x =0 u(x,0)=\ [[u_{L}, x < 0], [u_{R}, x > 0]] 3.3 Shock Path To determine the shock path we need to go back to the integral form of the conservation law which is still valid across the shock partial partial t integrate u dx from x_{L} to z_{R} = - (f(u_{R}) - f(u_{L})) HL n(z.t) u(x,t+t) The boundaries zy and x_{L} are taken sufficiently close to the shock, so that spatial variations of the solution away from the shock become unimportant. They are also taken sufficiently far apart from the shock so that the boundary will not interfere with the shock motion over a time interval öt. 10 - 1/(delta*t) * (u_{R} - u_{L}) * delta*x =- langle f(u R )-f(u L )) Shock speed N8 s= delta x delta t = (f(u_{R}) - f(u_{L}))/(u_{R} - u_{L}) equiv [f] [u] Rankine-Hugoniot jump condition Note 8 The Equal Area Rule One technique that is sometimes useful for determining the discontinuous solu-tion is to start with the solution constructed using characteristics. This solution may be multi-valued if characteristics cross. The multi-valued parts can be elim-inated by Inserting shocks at appropriate locations. The shock location can be determined by the equal area rule which is depicted in the figure below. The shock is placed so that the shaded areas cut off on either side have equal areas. 3.3.1 Example For our example, s = [f]/[u] = (0 - 1/2)/(0 - 1) = 1/2 x * = (1 + t)/2 u = 1 lambda_{x} = (1 + t)/2 u = 0 n = (1 - x)/(1 - 1) 3.3.2 Variable States If u is not constant at both sides of the shock, the jump condition still applies locally shock path is curved.

Description

Related Documents

Almost There!

Free up your schedule!

Take 5 seconds to unlock