MATH 676 Finite element methods in scientifjc computing Wolfgang - PowerPoint PPT Presentation

MATH 676 – Finite element methods in scientifjc computing Wolfgang Bangerth, T exas A&M University http://www.dealii.org/ Wolfgang Bangerth

Lecture 30.25: Time discretizations for advection- difgusion and other problems: IMEX, operator splitting, and other ideas http://www.dealii.org/ Wolfgang Bangerth

Explicit vs implicit time stepping Recall (lectures 26-28): ● Parabolic problems, e.g. heat equation: Implicit time stepping! ∂ u ∂ t −Δ u = f ● 2 nd order hyperbolic problems, e.g. wave equation: Explicit time stepping! 2 u ∂ 2 −Δ u = f ∂ t ● 1 st order hyperbolic problems, e.g. transport equation: Explicit time stepping! ∂ u ∂ t +⃗ β⋅∇ u = f http://www.dealii.org/ Wolfgang Bangerth

Questions Questions for this lecture: ● What do we do for problems that do not fall into these neat categories? ● What are common approaches? http://www.dealii.org/ Wolfgang Bangerth

Explicit vs implicit time stepping Example: What to do with advection-difgusion problems? ∂ u ∂ t + ⃗ β⋅∇ u − Δ u = f Suggests explicit Suggests implicit time stepping time stepping Note: Advection-difgusion equations describe processes where material/energy is transported and difguses (water, atmosphere, etc). Difgusion is often small. http://www.dealii.org/ Wolfgang Bangerth

IMEX schemes Example: What to do with advection-difgusion problems? Answer 1: Implicit/explicit ( IMEX ) schemes ● treat transport explicitly ● treat difgusion implicitly ∂ u ∂ t + ⃗ β⋅∇ u − Δ u = f n − 1 n − u u n = f ( k + ⃗ n − 1 − Δ u n = t n − t n − 1 ) β⋅∇ u n k n − k n Δ u n = u n ⃗ n − 1 − k n − 1 + k n f u β⋅∇ u http://www.dealii.org/ Wolfgang Bangerth

IMEX schemes Reformulation: Such schemes are often approximated in a way that separate the physical efgects: n − u n − 1 u n = f + ⃗ n − 1 − Δ u β⋅∇ u n k n − u n − 1 n − 1 n u adv u source − u + ⃗ n − 1 = 0 β⋅∇ u = f n − u k n k n n − 1 u diff n = 0 − Δ u diff k n n − u n − u n − 1 n − 1 n n − 1 n − u n − 1 = u adv + u diff + u source − u u k n k n k n k n http://www.dealii.org/ Wolfgang Bangerth

IMEX schemes Reformulation: Such schemes are often approximated in a way that separate the physical efgects: n − u n − 1 u n = f + ⃗ n − 1 − Δ u β⋅∇ u n k n = − k n ⃗ n − 1 n n f δ u adv β⋅∇ u δ u source = k n = k n ) n Δ( u n − 1 +δ u diff δ u diff n = u n − 1 + δ u adv n + δ u source n n u + δ u diff http://www.dealii.org/ Wolfgang Bangerth

IMEX schemes Reformulation: Such schemes are often approximated in a way that separate the physical efgects: n − 1 + δ u adv n + δ u source n = u n n + δ u diff u ● Computing increments can be done independently: – concurrently (in parallel) – by separate codes ● Source contribution may be included into the other solves ● Scheme can be generalized to higher order http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Example: What to do with advection-difgusion problems? Answer 2: Operator splitting schemes solve for one physical efgect after the other. With operator splitting, we can also ● treat transport explicitly ● treat difgusion implicitly Note: IMEX treats terms concurrently, operator splitting sequentially. http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Formulation: Operator splitting schemes separate the physical efgects: n − u n − 1 u n = f + ⃗ n − 1 − Δ u β⋅∇ u n k n − u n − 1 u adv + ⃗ n − 1 = 0 β⋅∇ u k n n − u adv n u diff n = 0 − Δ u diff n k n n − 1 − u diff u source = f k n This method is n = u source called “Lie” splitting n u http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Formulation: Operator splitting schemes separate the physical efgects. ● Computing 3 increments can be done – independently – by separate codes ● Source contribution may be included into the other solves ● The Lie scheme is only fjrst order in k n ● Scheme can be generalized to second order (“Strang splitting”) http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Example: Consider the reaction of 3 species A + B → C in a reactor. A simple model would be ● Solution variable: u ( x ,t )= { u A ( x ,t ) ,u B ( x,t ) ,u C ( x ,t ) } ● Equation: ∂⃗ u u = ⃗ ∂ t − Δ⃗ f (⃗ u ) ● Reaction terms: u ) = ( + k u A u B ) − ku A u B ⃗ f (⃗ − ku A u B http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Example: Consider the equation ∂⃗ u u = ⃗ ∂ t − Δ⃗ f (⃗ u ) Here: ● One term is a spatial process (difgusion, a PDE) ● One term is a local process (reaction, an ODE) ● We may have difgerent codes that are specialized in each process http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Example: Consider the equation ∂⃗ u u = ⃗ ∂ t − Δ⃗ f (⃗ u ) First order operator splitting (“Lie splitting”): ● First account for the efgect of one time step's worth of difgusion (implicit): * −⃗ n − 1 ⃗ u u * = 0 − Δ⃗ u n k ● Then account for one time step's worth of reactions (local ODE): ** ∂⃗ u * → ⃗ ∂ t = ⃗ ** ) , ⃗ ** ( t n − 1 )=⃗ n =⃗ ** ( t n ) f (⃗ u u u u u ● The order could of course be reversed. http://www.dealii.org/ Wolfgang Bangerth

Operator splitting schemes Example: Consider the equation ∂⃗ u u = ⃗ ∂ t − Δ⃗ f (⃗ u ) Second order operator splitting (“Strang splitting”): ● Half difgusion step: n − 1 * −⃗ ⃗ u u * = 0 − Δ⃗ u n / 2 k ● Full reaction step: ** ∂⃗ u * → solve for ⃗ ∂ t = ⃗ ** ) , ⃗ ** ( t n − 1 )=⃗ ** ( t n ) f (⃗ u u u u ● Half difgusion step: n −⃗ ** ( t n ) ⃗ u u n = 0 − Δ⃗ u k n / 2 ● The order of sub-steps can be reversed. http://www.dealii.org/ Wolfgang Bangerth

More accuracy Background, part 1: Both IMEX and Operator Splitting schemes need to discretize the time derivative ∂⃗ u ∂ t This can be done in many ways, for example: ● Simplest approximation (Euler, BDF-1, ...) n − u n − 1 ∂ u ∂ t ≈ u k ● BDF-2 3 n − 1 + 1 n − 2 n − 2 u 2 u 2 u ∂ u ∂ t ≈ k http://www.dealii.org/ Wolfgang Bangerth

More accuracy Background, part 2: We need to approximate explicit terms in equations such as ∂ u ∂ t + ⃗ β⋅∇ u − Δ u = f This can be done in many ways, for example: ● Explicit Euler n − u n − 1 u n = f + ⃗ n − 1 − Δ u β⋅∇ u n k ● T wo-step (explicit) extrapolation β⋅∇ ( u n − 1 + k n u ) − Δ u n = f n − 1 n − 1 − u n − 2 n − u u + ⃗ n n − 1 k k http://www.dealii.org/ Wolfgang Bangerth

Summary Many important, time-dependent equations are not purely ● parabolic ● Hyperbolic. For these equations, one often wants to treat ● some terms explicitly ● some terms implicitly ● treat difgerent physical efgects separately. There are many ways of doing this (e.g., IMEX, Operator Splitting) and many variations to achieve higher order accuracy. http://www.dealii.org/ Wolfgang Bangerth

MATH 676 – Finite element methods in scientifjc computing Wolfgang Bangerth, T exas A&M University http://www.dealii.org/ Wolfgang Bangerth