University:
Boston University
Course:
CAS MA 511 | Introduction to Analysis I
Academic year:

2022
Views:

362

Pages:

18
Author:

Patricia West

The Riemann problem shallow water wave systems Plan Part I : Discussion of the Riemann solution for onedimensional linear and nonlinear shallow water wave equations Part II : Approximate Riemann solvers in GeoClaw; discussion of accuracy, and extensions to higher dimensions; f-wave approach to well-balancing. Part III : Adaptive mesh refinement (AMR). Shallow water wave equations The shallow water wave equations, given by is an example of a system of equations written in conservative form. More generally, we can write PDEs in conservative form as qt + f (q)x = 0 These are typically derived from conservation laws for mass, momentum, energy, species, and so on. GeoClaw • Based on solving the conservative form of the shallow water wave equations using a finite volume method. • At the heart of many finite volume methods is a Riemann solver which is used to compute numerical fluxes. • In GeoClaw, these are stored in files like rpn2_geo.f and rpt2_geo.f Finite volume method Assume a conservation law of the form qt + f (q)x = 0 Define cell averages over the interval Ci = [xi Qni 1 = x 1/2 , xi+1/2 ] q(x, tn ) dx Ci How does the average evolve? d dt d f (q(x, t)) dx q(x, t) dx = Ci Ci dx = f (q(xi 1/2 , t)) f (q(xi+1/2 , t)) Finite volume method Evolution of the cell average value : d dt Z q(x, t) dx = f (q(xi 1/2 , t)) f (q(xi+1/2 , t)) Ci Integrate in time ⇤ q(x, tn+1 ) dx = Ci + ⇤ q(x, tn ) dx Ci ⇤ tn+1 tn f (q(xi 1/2 , t)) ⇥ f (q(xi+1/2 , t)) dt Finite volume method Using numerical fluxes, we use the update formula : n+1 Qi = n Qi t n Fi+1/2 x Written as this form resembles the conservation law : n Fi 1/2 ⇥ Numerical fluxes We want to approximate the numerical flux. Z tn+1 1 n Fi 1/2 ⇡ f (q(xi 1/2 , t)) dt t tn For an explicit time stepping scheme, we try to find formulas for the flux of the form n Fi 1/2 Qi 2 Qi n n F(Qi , Qi 1 ) = 1 Qi xi 1/2 Qi+1 1d Riemann problem At each cell interface, solve the hyperbolic problem with special initial data, i.e. qt + f (q)x = 0 subject to q(x, 0) = ⇢ Qi Qi Qi 1 x < xi x > xx 1 t=0 Qi xi 1/2 1/2 1/2 1d Riemann problem Qi t=0 1 Qi xi t>0 Qi 1/2 q⇤ 1 Qi xi 1/2 Solve the conservation law with piecewise constant initial data 1d Riemann problem q t>0 Qi ⇤ 1 Qi xi 1/2 Numerical flux at cell interface is then approximated by Fi 1/2 = f (q ) ⇤ This is the classical Godunov approach for solving hyperbolic conservation laws. • Resolves shocks and rarefactions Conservation? Integrating over entire domain, we have d dt Z Z xb q(x, t) dx = xa xb (f (q))x dx = f (q(xa , t)) f (q(xb , t)) xa Discrete case M X Qn+1 i = i=1 M X Qni i=1 = M X i=1 Qni M X t Fi+1/2 x i=1 t FM +1/2 x Fi 1/2 F1/2 Quantities are conserved up to fluxes at domain boundaries. Scalar advection Consider the constant initial value problem qt + ūqx = 0 It is easy to verify that solves the initial value problem. Scalar advection We can describe the problem in terms of how the solution behaves along curves in the x-t plane. We might look for curves solution is constant or Then we would get along which the Characteristic curves But this is true only if or Solution is constant along characteristic curves. For , Characteristic curves The solution can be traced back along characteristics. That is, can be found by determining the from which the solution propagated. Solve or Scalar advection Consider the scalar advection equation : qt + ūqx = 0 The solution travels along characteristic rays in the (x,t) plane given by . For u < 0 : t=1 t=0 Riemann problem for scalar advection qt + ūqx = 0 subject to particular initial conditions q(x, 0) = q qr t q Solution : qr x<0 x>0