Convection-Dominated Problems Chapter 10 Nonlinear Convection-Dominated Problems

10.1 Burgers’ Equation One-dimensional Burgers’ equation Conservative form

Inviscid Burgers’ Equation One-dimensional inviscid Burgers’ equation Larger values of convect faster and overtake slower Multi-valued solution may occur Postulate a shock to allow the development of discontinuous solutions

Inviscid Burgers’ Equation Formation of multi-valued solution The nonlinearity allows discontinuous solutions to develop Shock-fitting b t = t1 t = t0 t = t2 shock a

Viscous Burgers’ Equation Viscous term reduces the amplitude in high gradient regions Prevents multi-valued solutions from developing (second derivative increases faster than first derivative) t = t0 t = t1 t = t2

10.1.2 Explicit Schemes FTCS scheme (non-conservative) FTCS (conservative form)

Explicit Schemes Four-point Upwind Scheme Truncation errors O(x2) if q  0.5 O(x3) if q = 0.5

Lax-Wendroff Scheme Inviscid Burgers’ equation for unsteady one-dimensional shock flows Replace temporal derivative by equivalent spatial derivative (more complicated for nonlinear case) Chain rule

Lax-Wendroff Scheme Central-difference discretization For Burgers’ equation

Lax-Wendroff Scheme Temporal derivative Inviscid Burgers’ equation Rearrange

Lax-Wendroff Scheme Linear pure convection equation Nonlinear - inviscid Burgers’ equation Equivalent two-stage algorithm (more economical)

Burgers’ Equation Thommen’s extension of Lax-Wendroff scheme for viscous flow problems Error in textbook Stability limit

10.1.3 Implicit Schemes Burgers’ equation (viscous) Crank-Nicolson implicit formulation Thomas algorithm for tridiagonal matrices cannot be used directly due to the appearance of nonlinear implicit term Use Taylor series expansion of at nth time- level to convert to tridiagonal form

Crank-Nicolson Scheme Taylor-series expansion (linearlization of F) Linear tridiagonal system (in terms of u or u)

Crank-Nicolson Scheme Thomas algorithm The matrix coefficients must be reevaluated at every time step (to recover nonlinearity of the equation) Truncation error O(t2, x2) Unconditionally stable in Von Neumann sense (linear)

Generalized Crank-Nicolson Mass operator and four-point upwind Truncation error O(t2, x2)

Generalized Crank-Nicolson Quadridiagonal system of equations – can be solved using generalized Thomas algorithm

Artificial Dissipation Crank-Nicolson with additional dissipation For small values of viscosity (high-Re), it is desirable to add some artificial dissipation Modified Crank-Nicolson Choose a empirically

10.1.4 BURG: Numerical Comparison Propagation of a shock wave governed by viscous Burgers’ equation Exact solution

Burgers’ Equation

BURG: Numerical Comparison ME = 1, FTCS scheme ME = 2, two-stage Lax-Wendroff scheme ME = 3, Explicit four-point upwind scheme ME = 4, Crank-Nicolson (CN-FDM):  = 0, q = 0 ME = 4, Crank-Nicolson (CN-FEM):  = 1/6, q = 0 ME = 4, Crank-Nicolson, Mass Operator (CN-MO):  = 1/12, q = 0 ME = 4, Crank-Nicolson, 4-pt. Upwind (CN-4PU):  = 0, q = 0.5 ME = 5, Crank-Nicolson plus additional dissipation Note: Optimum  and q (locally freezing nonlinear coefficients)

Burgers’ Equation

Burgers’ Equation Propagating Shock Solution Rcell = 1.0, C = 0.25

Burgers’ Equation: Propagating Shock Rcell = 3.33, C = 1.0 Rcell = 100, C = 1.0

Velocity distribution at t = 2.0; Rcell = 100

10.2 Systems of Equations Continuity equation Momentum equations Energy equation Equation of state (compressible flows) Turbulent kinetic energy equation Rate of turbulent energy dissipation equation Reynolds stresses equations Multiphase flows Chemical reactions

Systems of Equations 1D unsteady compressible inviscid flow Continuity equation, x-momentum equation, energy equation

Two-Stage Lax-Wendroff Single equation System of equations

Lax-Wendroff Scheme with Artificial Viscosity Continuity equation X-momentum equation Energy equation

Crank-Nicolson Scheme System of equations Linearization 33 block tridiagonal system (solved by block Thomas algorithm) (33 matrix)

Crank-Nicolson Scheme Use Von Neumann analysis for the linearized equation Amplification matrix Numerical Stability

10.3 Group Finite Element Method Conventional finite element method introduces a separation approximate solution (trial function, interpolation function) for each dependent variable Galerkin method produces large numbers of products of nodal values of dependent variables, particularly from the nonlinear convective terms Inefficient, time-consuming Group finite element formulation is effective in dealing with convective nonlinearities

Group Finite Element Method Group finite element formulation The equations are cast in conservative form A single approximation solution is used for the group of terms in the differential terms (i.e., approximate F directly instead of the nonlinear convective term uu/x) One-dimensional Group Formulation

Group Finite Element Method One-dimensional Group Formulation Conventional finite element Conservative form Non-conservative form

One-dimensional Burgers’ equation Conventional and group FEMs

10.4 2D Burgers’ Equation Two-dimensional Burgers’ equation Equivalent to 2D momentum equations for incompressible laminar flow with zero pressure gradient

2D Burgers’ Equation Exact solution Use Cole-Hopf transformation Transform the 2D Burgers’ equation into one single equation – 2D diffusion equation

2D Burgers’ Equation Steady 2D Burgers’ equation Exact solution

Exact solution for 2D Burgers’ equation

2D Burgers’ Equation – Exact u a1= a2= 1.3*1013, a3= a4= 0, a5 = 1,  = 25, x0 =1,  = 0.04

2D Burgers’ Equation – Exact v a1= a2= 1.3*1013, a3= a4= 0, a5 = 1,  = 25, x0 =1,  = 0.04

Multidimensional Group FEM Two-dimensional Burgers’ equation Approximate solutions for (u,v), and groups (u2, uv, v2) and the components of S For example (bilinear for rectangular elements)

Galerkin Finite Element Linear (Chapter 9) Nonlinear (Group FE formulation) The equations are treated as linear at the level at which the discretization take place (but indeterminate) Substitution for the nodal groups in terms of the unknown nodal variables introduces the nonlinearity but also makes the system determinate

Split Schemes Two-dimensional Burgers’ equations Similar to those used in Chapters 8 and 9 Additional complication due to nonlinearity Generalized FEM/FEM with mass operators Mx and My

Pseudo-Transient Formulation Use pseudo-transient formulation (sect 6.4) for steady-state solution For steady-state problems, unsteady formulation provides an equivalent underrelaxation parameter for steady iterative schemes For steady-state solutions, it is desirable to use a simple time discretization (such as two-level fully implicit scheme with  = 1) to simplify the formulation

Pseudo-Transient Formulation Two-level fully implicit scheme ( = 1) Linearize the nonlinear terms F, G, and S in (RHS)n+1

Pseudo-Transient Formulation Linearization (Jacobian matrices A, B, C) Approximate Factorization

Pseudo-Transient Formulation Further simplification to reduce CPU time Use the same left-hand-side for each scalar component Perform only one factorization (BANFAC) for different components Does not affect the steady-state solution since (RHS)n = 0 in the steady state limit

TWBURG: Numerical Solution Two-dimensional Burgers’ equations Steady state solution with the following split algorithm Solution domain 1 x  1 , 0  y  ymax , ymax= /6 Use exact solution for the boundary conditions Initial conditions obtained from linear interpolation of the boundary condition in the x-direction

Computer Program - TWBURG

Approximate Factorization

Error Distributions at y/ymax = 0.4