A positivity-preserving flux-corrected transport scheme for solving - PowerPoint PPT Presentation

A positivity-preserving flux-corrected transport scheme for solving scalar conservation law problems Joshua E. Hansel 1 Jean C. Ragusa 1 Jean-Luc Guermond 2 1 Department of Nuclear Engineering Texas A&M University 2 Department of Mathematics Texas A&M University deal.II Workshop, Summer 2015

Motivation Introduction Motivation Objectives Weak solutions to conservation law problems in general Outline are not unique; thus solution via CFEM prone to Methodology Formulation unphysical oscillations: Time Discretization 1.2 FCT Scheme Exact Galerkin Overview Low-Order 1 Scheme High-Order Scheme 0.8 FCT Scheme Results 0.6 Solution Conclusions 0.4 0.2 0 -0.2 0 0.2 0.4 0.6 0.8 1 x 2/ 29

Objectives Introduction Motivation Objectives Outline The objectives of the research are the following: Methodology Accurately solve conservation law problems using the Formulation Time continuous finite element method (CFEM). Discretization FCT Scheme Overview Scheme to be presented is 2nd order-accurate in space Low-Order Scheme (for smooth problems). High-Order Scheme Prevent spurious oscillations . FCT Scheme Scheme to be presented is not proven to be completely Results immune to any spurious oscillations but shows good Conclusions results in practice. Prevent negativities for physically non-negative quantities. Scheme to be presented is guaranteed to be positivity-preserving. 3/ 29

Outline Introduction Motivation Objectives Outline Methodology Formulation Time Presentation of scheme for simple case Discretization FCT Scheme Overview Problem formulation Low-Order Scheme Monotone low-order scheme High-Order Scheme High-order entropy viscosity scheme FCT Scheme FCT scheme Results Conclusions Results Conclusions 4/ 29

Conservation Law Models Introduction Motivation Guermond has addressed these objectives for general Objectives Outline nonlinear scalar conservation laws using explicit temporal Methodology discretizations: Formulation Time ∂ u Discretization ∂ t + ∇ · f ( u ) = 0 FCT Scheme Overview Low-Order Scheme Common examples: High-Order Scheme FCT Scheme f ( u ) = u v Linear advection equation Results f ( u ) = 1 2 u 2 v Burgers equation Conclusions We extend these techniques to include a reaction term and source term and to use implicit and steady-state temporal discretizations: ∂ u ∂ t + ∇ · f ( u ) + σ u = q 5/ 29

Problem Formulation Introduction Scalar linear conservation law model: Motivation Objectives Outline ∂ u ∂ t + ∇ · ( v u ( x , t )) + σ ( x ) u ( x , t ) = q ( x , t ) (1) Methodology Formulation Time Discretization σ ( x ) ≥ 0 , q ( x , t ) ≥ 0 FCT Scheme Overview Low-Order Define problem by providing initial conditions and some Scheme High-Order Scheme boundary condition, such as Dirichlet: FCT Scheme Results u ( x , 0) = u 0 ( x ) ∀ x ∈ D (2) Conclusions u ( x , t ) = u inc ( x ) ∀ x ∈ ∂ D inc (3) CFEM solution: N � ϕ j ( x ) ∈ P 1 u h ( x , t ) = U j ( t ) ϕ j ( x ) , (4) h j =1 6/ 29

Time Discretization Introduction Motivation Simplest time discretization is forward Euler (FE), which Objectives Outline gives the discrete system Methodology Formulation M C U n +1 − U n Time Discretization + AU n = b n (5) FCT Scheme Overview ∆ t Low-Order Scheme High-Order � Scheme M C i , j ≡ ϕ i ( x ) ϕ j ( x ) d x (6) FCT Scheme Results S i , j Conclusions � A i , j ≡ ( v · ∇ ϕ j ( x ) + σ ( x ) ϕ j ( x )) ϕ i ( x ) d x (7) S i , j � b n q ( x , t n ) ϕ i ( x ) d x i ≡ (8) S i 7/ 29

Flux Corrected Transport (FCT) Scheme Introduction Introduction Motivation Objectives Initially developed in 1973 for finite difference Outline Methodology discretizations of transport/conservation law problems and Formulation Time recently applied to finite element method. Discretization FCT Scheme Overview Works by adding conservative fluxes to satisfy physical Low-Order Scheme bounds on the solution. High-Order Scheme FCT Scheme Employs a high-order scheme and a low-order, monotone Results scheme. Conclusions Defines a correction , or antidiffusion , flux, which when added to the low-order scheme, produces the high-order scheme solution. Limits this correction flux to enforce the physical bounds imposed. 8/ 29

Low-Order Scheme Definition Introduction Motivation To get the low-order scheme, one does the following: Objectives Lumps the mass matrix: M C → M L . Outline Methodology Adds a low-order diffusion operator: A → A + D L . Formulation Time This gives the following, where U L , n +1 is the low-order Discretization FCT Scheme Overview solution: Low-Order Scheme High-Order M L U L , n +1 − U n Scheme + ( A + D L ) U n = b n (9) FCT Scheme ∆ t Results Conclusions The diffusion matrix D L is assembled elementwise, where K denotes an element, using a local bilinear form b K and a local low-order viscosity ν L K : D L � ν L i , j = K b K ( ϕ j , ϕ i ) (10) K ⊂ S i , j 9/ 29

Low-Order Scheme Local Bilinear Form Introduction Motivation The local bilinear form is defined as follows, where | K | Objectives Outline denotes the volume of element K , I ( K ) is the set of Methodology indices corresponding to degrees of freedom with Formulation Time nonempty support on K , and n K is the cardinality of this Discretization FCT Scheme set. Overview Low-Order Scheme High-Order 1  − n K − 1 | K | i � = j , i , j ∈ I ( K ) Scheme  FCT Scheme b K ( ϕ j , ϕ i ) ≡ (11) | K | i = j , i , j ∈ I ( K ) Results 0 i / ∈ I ( K ) | j / ∈ I ( K )  Conclusions Some properties that result from this definition: � b K ( ϕ j , ϕ i ) = 0 (12) j b K ( ϕ i , ϕ i ) > 0 (13) 10/ 29

Low-Order Scheme Low-Order Viscosity Introduction Motivation Objectives Outline The low-order viscosity is defined as Methodology Formulation max(0 , A i , j ) Time ν L K ≡ max (14) Discretization − � b T ( ϕ j , ϕ i ) FCT Scheme i � = j ∈I ( K ) Overview Low-Order T ⊂ S i , j Scheme High-Order Scheme This definition is designed to be the smallest number such FCT Scheme Results that the following is guaranteed: Conclusions D L i , j ≤ − A i , j , j � = i (15) This is used to guarantee that the low-order steady-state matrix A L = A + D L is an M-matrix, i.e., a monotone matrix: A L U ≥ 0 ⇒ U ≥ 0. 11/ 29

Low-Order Scheme Discrete Maximum Principle Introduction In addition to guaranteeing monotonicity and positivity, Motivation Objectives the low-order viscous terms guarantee the following Outline Methodology discrete maximum principle (DMP), where Formulation max Time U n j ∈I ( Si ) U n min , i = j : min Discretization max FCT Scheme Overview Low-Order ≤ U L , n +1 Scheme W − ≤ W + ∀ i (16) High-Order i i i Scheme FCT Scheme   Results  1 − ∆ t  + ∆ t � W ± ≡ U n A L b n (17) Conclusions max i , j i i min , i M L M L i , i i , i j For example, when there is no reaction term or source term, this reduces to the following DMP, which implies the scheme is local extremum diminishing (LED): min , i ≤ U L , n +1 U n ≤ U n ∀ i (18) max , i i 12/ 29

Low-Order Scheme Getting Nonzero Row Entries { A i , j : A i , j � = 0 , j = 1 . . . N } Introduction Motivation Objectives Outline void get_matrix_row ( const SparseMatrix <double > &matrix , Methodology const unsigned int &i, Formulation std :: vector <double > &row_values , Time std :: vector <unsigned int > &row_indices , Discretization unsigned int &n_col) FCT Scheme Overview { Low-Order // get first and one -past -end iterator for row Scheme SparseMatrix <double >:: const_iterator it = matrix.begin(i); High-Order SparseMatrix <double >:: const_iterator it_end = matrix.end(i); Scheme FCT Scheme // determine number of entries in row and resize vectors accordingly Results n_col = it_end - it; row_values .resize(n_col ); Conclusions row_indices .resize(n_col ); // loop over columns in row for (unsigned int k = 0; it != it_end; ++it , ++k) { row_values [k] = it ->value (); // get A(i,j) row_indices [k] = it ->column (); // get j } } 13/ 29

Low-Order Scheme Results Example Introduction 1.2 Exact Motivation Galerkin Objectives Low-Order Outline DMP-min 1 DMP-max Methodology Formulation Time Discretization 0.8 FCT Scheme Overview Low-Order Scheme 0.6 High-Order Scheme Solution FCT Scheme Results 0.4 Conclusions 0.2 0 -0.2 0 0.2 0.4 0.6 0.8 1 x 14/ 29

Entropy Viscosity Scheme Introduction Introduction The standard Galerkin CFEM weak solution is not unique. Motivation Objectives Even with FCT, it would not necessarily converge to the Outline Methodology correct, physical weak solution, i.e., the entropy solution. Formulation To converge to the entropy solution, one must ensure that Time Discretization FCT Scheme an entropy inequality is satisfied: Overview Low-Order Scheme R ( u ) ≡ ∂η ( u ) High-Order + ∇ · f η ( u ) ≤ 0 (19) Scheme ∂ t FCT Scheme Results for any convex entropy η ( u ) and corresponding entropy Conclusions flux f η ( u ). This entropy residual R ( u ) measures entropy production; where it is positive, the inequality is violated, so the residual should be decreased somehow. To enforce the inequality, the entropy viscosity method adds viscosity in proportion to local entropy production, thus decreasing local entropy. 15/ 29