Tutorial Part III (Time-dependent Problems)

This chapter you will learn how to solve systems of equations and time-dependent problems.

• Time-Dependent Problems (01-cathedral-ie)
  • Model problem
  • Weak formulation
  • Defining weak forms and accessing external functions
  • Setting initial condition
  • Registering external functions in weak forms
  • Initializing the discrete problem
  • Assembling and the ‘rhs_only’ flag
• Time-Integration with Runge-Kutta Methods (02-cathedral-rk)
  • Runge-Kutta methods and Butcher’s tables
  • Butcher’s tables currently available in Hermes
  • Model problem
  • Weak formulation
  • Defining weak forms
  • Selecting a Butcher’s table
  • Registering weak forms
  • Setting initial condition
  • Initializing the discrete problem
  • Time-stepping loop
• Nonlinear Time-Dependent Problem (03-newton-heat-ie)
  • Model problem
  • Weak formulation
  • Jacobian matrix and residual vector
  • Time stepping loop
• Flame Propagation Problem (06-flame)
  • Model problem
  • Second-order BDF formula for time integration
  • Initial conditions
  • Using Filters
  • Registering weak forms
  • Reinitialization of Filters
  • Visualization of a Filter
  • Sample results
• Navier-Stokes Equations (07-navier-stokes)
  • Model problem
  • Role of pressure in Navier-Stokes equations
  • Defining spaces
  • Defining projection norms
  • Calculating initial coefficient vector for the Newton’s method
  • Time stepping
  • Sample results
• Gross-Pitaevski Equation (08-gross-pitaevski)
  • Model problem
  • Complex-valued Jacobian matrix and residual vector
  • Registration of weak forms
  • Time stepping loop
  • Sample results