A Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

A Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations Hao Li Math Dept, Purdue Univeristy Ocean University of China, December, 2017 Joint work with Shusen Xie (Ocean University of China) and Xiangxiong Zhang (Purdue University) 1 / 42

Overview 1 Inroduction Bound Preserving Conservative Scheme Time Discretization Monotone Scheme 2 Compact Finite Difference Weak Monotonicity 3 Further Extension Different Boundary Conditions Higher Order Scheme Non-uniform Grids 4 Numerical Test 2 / 42

Stability: Compressible Euler Equations in Gas Dynamics ρ m E t + m ρu 2 + p (E + p)u x = 0, with m = ρu, E = 1 2 ρu2 + ρe, p = (γ 1)ρe The speed of sound is given by c = γp/ρ and the three eigenvalues of the Jacobian are u, u ± c If either ρ < 0 or p < 0, then the sound speed is imaginary and the system is no longer hyperbolic Thus the initial value problem is ill-posed This is why it is computationally unstable 3 / 42

Bound Preserving for Scalar Conservation Laws Consider the initial value problem u t + F(u) = 0, u(x, 0) = u 0 (x), x R n for which the unique entropy solution u(x, t) satisfies min x In particular, u(x, t 0 ) u(x, t) max u(x, t 0 ), t t 0 Maximum Principle x min x u 0 (x) = m u(x, t) M = max u 0 (x) x It is also a desired property for numerical solutions due to Bound Preserving 1 Physical meaning: vehicle density (traffic flow), mass percentage (pollutant transport), probability distribution (Boltzmann equation) and etc 2 Stability for systems: positivity of density and pressure (gas dynamics), water height (shallow water equations), particle density for describing electrical discharges (a convection-dominated system) and etc For numerical schemes, this is a completely DIFFERENT problem from discrete maximum principle in solving elliptic equations 4 / 42

Scalar Equations IVP: u t + f(u) x = 0, u(x, 0) = u 0 (x) Maximum Principle (Bound Preserving): u(x, t) [m, M] where m = min u 0 (x), M = min u 0 (x) For finite difference, any scheme satisfying min u n j u n+1 j max u n j j j at most first order accurate can be Harten s Counter Example: consider u t + u x = 0, u(x, 0) = sin x Put the grids in a way such that x = π 2 is in the middle of two grid points

Bound-Preserving Schemes 1 First order monotone schemes 2 FD/FV schemes satisfying min u n j u n+1 j j max u n j : can have any formal j order of accuracy in the monotone region but are only first order accurate around the extrema Eg, Conventional total-variation-diminishing (TVD) schemes High Resolution schemes such as the MUSCL scheme 3 FV schemes satisfying min x u n (x) u n+1 (x) max u n (x): x R Sanders, 1988: a third order finite volume scheme for 1D XZ and Shu, 2009: higher order (up to 6th) extension of Sanders scheme Liu and Osher, 1996: a third order FV scheme for 1D (can be proven bound-preserving only for linear equations) Noelle, 1998; Kurganov and Petrova, 2001: 2D generalization of Liu and Osher All schemes in this category use the exact time evolution 8 / 42

Bound-Preserving Schemes Practical/popular high order schemes are NOT bound-preserving It was unknown previously how to construct a high order bound-preserving scheme for 2D nonlinear equations 9 / 42

Conservative Eulerian Schemes Conservative Schemes: The scheme must have the following form: U n+1 j = U n j t [ˆfj+ 1 x j 2 Global Conservation: u n+1 j x j = uj n x j j j We insist on using conservative schemes: ˆf j 1 2 1 Lax-Wendroff Theorem: If converging (as mesh sizes go to zero), the converged solution of a conservative scheme is a weak solution 2 The shock location will be wrong if the conservation is violated 3 If a scheme is conservative and positivity preserving, then we have L1-stability: j un+1 j x j = j un+1 j x j = j un j x j = j un j x j In Euler equations, if density and pressure are positive, then we have L1-stability for density and total energy Crude replacement of negative values by positive ones is simply unacceptable and unstable because it destroys the local conservation ] 10 / 42

Objective We want to construct a scheme which is 1 genuinely high order accurate (at least third order) 2 conservative 3 positivity/bound preserving 4 Practical concern: cost effective, multi-dimensions, unstructured meshes, parallelizability and etc Why high order schemes? For smooth solutions, the computational cost of higher order schemes is smaller to reach the same accuracy Near shocks, the error in L norm of any scheme will at most half order For a lot of problems: accuracy is still high order away from discontinuities Motivation: nonlinear stability is one of the main reasons why high order schemes have not been widely used for more real world problems 11 / 42

Time Discretization: SSP Runge-Kutta or Multi-Step Method High order strong stability preserving (SSP) Runge-Kutta or multi-step method is a convex combination of several forward Euler schemes Eg, the 5 stages 4th order SSP Runge-Kutta method for solving u t = F(u) is given by u (1) = u n + 0391752226571890 tf(u n ) u (2) = 0444370493651235u n + 0555629506348765u (1) +0368410593050371 tf(u (1) ) u (3) = 0620101851488403u n + 0379898148511597u (2) +0251891774271694 tf(u (2) ) u (4) = 0178079954393132u n + 0821920045606868u (3) +0544974750228521 tf(u (3) ) u n+1 = 0517231671970585u (2) + 0096059710526147u (3) +0063692468666290 tf(u (3) ) + 0386708617503269u (4) +0226007483236906 tf(u (4) ) This method was numerically found to be optimal 12 / 42

Time Discretization: SSP Runge-Kutta or Multi-Step Method If the forward Euler is bound-preserving, then so is the high order Runge-Kutta/Multi-Step SSP time discretization has been often used to construct positivity preserving schemes 13 / 42

First Order Monotone Schemes Let λ = t x, a monotone scheme for u t + f(u) x = 0 is given by ] u n+1 j = u n n j λ [ f(u j, u n j+1) f(u n j 1, u n j ) = H(u n j 1, u n j, u n j+1) where the numerical flux f(, ) is monotonically increasing wrt the first variable and decreasing wrt the second variable Eg, the Lax-Friedrichs flux 1 f(u, v) = (f(u) + f(v) α(v u)), α = max f (u) 2 u If m u n j M for all j, then u n+1 j = H(u n j 1, u n j, u n j+1) = H(,, ) (1) implies m = H(m, m, m) u n+1 j H(M, M, M) = M 14 / 42

Fourth Order Compact Finite Difference Standard centered finite difference: u i = u i+1 u i 1 2 x + O( x 2 ), u i Fourth order compact finite difference: = u i+1 2u i + u i 1 x 2 + O( x 2 ) 1 6 u i+1 + 4 6 u i + 1 6 u i 1 = u i+1 u i 1 + O( x 4 ) 2 x 1 12 u i+1 + 5 6 u i + 1 12 u i 1 = u i+1 2u i + u i 1 x 2 + O( x 4 ) A tridiagonal system needs to be solved: 1 6 4 1 1 1 4 1 1 4 1 1 4 1 1 1 4 u 1 u 2 u 3 u N 1 u N = 1 2 x ie Wu = D x u 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 u 1 u 2 u 3 u N 1 u N, 15 / 42

The Weighting Operator for Convection If we regard W as an operator mapping a vector to another vector, then (Wu) j = 1 6 u j+1 + 4 6 u j + 1 6 u j 1, which happens to be the Simpson s rule (or 3-point Gauss-Lobatto Rule) in quadrature x j 1 x j x j+1 Locally, for each interval [x j 1, x j+1 ], there exists a cubic polynomial p j (x), obtained through interpolation at x j 1, x j, x j+1, x j+2 (or x j 2, x j 1, x j, x j+1 )

The Weighting Operator for Convection If we regard W as an operator mapping a vector to another vector, then (Wu) j = 1 6 u j+1 + 4 6 u j + 1 6 u j 1, which happens to be the Simpson s rule (or 3-point Gauss-Lobatto Rule) in quadrature x j 1 x j x j+1 x j+2 Locally, for each interval [x j 1, x j+1 ], there exists a cubic polynomial p j (x), obtained through interpolation at x j 1, x j, x j+1, x j+2 (or x j 2, x j 1, x j, x j+1 )

The Weighting Operator for Convection If we regard W as an operator mapping a vector to another vector, then (Wu) j = 1 6 u j+1 + 4 6 u j + 1 6 u j 1, which happens to be the Simpson s rule (or 3-point Gauss-Lobatto Rule) in quadrature x j 1 x j x j+1 x j+2 Locally, for each interval [x j 1, x j+1 ], there exists a cubic polynomial p j (x), obtained through interpolation at x j 1, x j, x j+1, x j+2 (or x j 2, x j 1, x j, x j+1 ) 18 / 42

The Fourth Order Compact Finite Difference Scheme for Convection Let u i = (Wu) i = 1 6 u i 1 + 4 6 u i + 1 6 u i+1 The fourth order compact finite difference for u t + f(u) x = 0 can be written as u n+1 i = u n i t 1 x 2 [f(un i+1) f(u n i 1)] = u n i t x (ˆf n ˆf n ), i+ 1 i 1 2 2 ˆfi+ 1 2 = 1 2 (f(u i+1) + f(u i )) The weak monotonicity holds under the CFL constraint λ max u f (u) 1 3 : u n+1 i = 1 6 un i 1 + 4 6 un i + 1 6 un i+1 + 1 2 λ[f(un i+1) f(u n i 1)] = 1 6 [u i 1 3λf(u n i 1)] + 1 6 [un i+1 + 3λf(u n i+1)] + 4 6 un i = H(u n i 1, u n i, u n i+1) = H(,, ) Thus m u n i M implies m = H(m, m, m) u n+1 i H(M, M, M) = M 19 / 42

How to enforce bounds? Consider enforcing the positivity (general lower/upper bounds can be similarly treated) Given point values u i, if 1 6 u i 1 + 4 6 u i + 1 6 u i+1 0 for any i, then the following hold: For any three consecutive points, at least one point value is non-negative If u i < 0, local average is non-negative: 1 3 u i 1 + 1 3 u i + 1 3 u i+1 0 This means that it is possible to eliminate the undershoot without changing 1 3 u i 1 + 1 3 u i + 1 3 u i+1 If u i < 0, then either u i + 1 2 u i+1 0 or u i + 1 2 u i 1 0 So a simple local limiter can enforce the positivity ( or general lower/upper bounds): if u i+1 u i 1, set u i+1 u i+1 + u i, u i 0, if u i+1 < u i 1, set u i 1 u i 1 + u i, u i 0 20 / 42

2D Convection Let u i,j denote point values on a 2D uniform grid Define two weighting operators: (W x u) i,j = 1 6 u i 1,j + 4 6 u i,j + 1 6 u i+1,j, (W y u) i,j = 1 6 u i,j 1 + 4 6 u i,j + 1 6 u i,j+1 Let u denote W x W y u The fourth order compact finite difference for u t + f(u) x + g(u) y = 0 can be written as u n+1 i,j = u n i,j + t 1 x 2 W y[f(u n i+1,j) f(u n i 1,j)] + t 1 y 2 W x[g(u n i,j+1) f(u n i,j 1)] has weak monotonicity thus u n i,j A dimension by dimension fashion 2D limiter: u n+1 i,j 0 implies un+1 i,j 0 Introduce a new variable v = W x u So u = W y v Given u 0, apply 1D limiter in y-direction on v to enforce v 0 Given v 0, apply 1D limiter in x-direction on u to enforce u 0 21 / 42

Diffusion 1 12 (u i+1 + 10u i + u i 1) = u i+1 2u i + u i 1 x 2 + O( x 4 ) Let ũ i = (Wu) i = 1 12 (u i 1 + 10u i + u i+1 ) The fourth order compact finite difference for u t = g(u) xx can be written as ũ n+1 i = ũ n i + t x 2 [g(un i+1) 2g(u n i ) + g(u n i 1)], Assuming g (u) 0 The weak monotonicity holds under the CFL constraint t x max 2 u f (u) 1 6 The same 1D limiter can be used for enforcing bounds Remarks For approximating the 2D Laplacian, the fourth order compact finite difference scheme is also known as the 9-point discrete Laplacian (the second order centered difference is 5-point Laplacian) For solving heat equation, Crank-Nicolson time discretization and the 9-point Laplacian give an M-matrix, thus it can preserve bounds/positivity However, such a result cannot be easily extended to nonlinear cases 22 / 42

Convection-Diffusion For the equation u t + f(u) x = g(u) xx, define two weighting operators: (W 1 u) j = 1 6 (u j+1 + 4u j + u j 1 ), (W 2 u) j = 1 12 (u j+1 + 10u j + u j 1 ) Let ū i = (W 1 W 2 u) i = (W 2 W 1 u) i, then the scheme is ū n+1 j = ū n j λ 1 2 W 2[f(u n j+1) f(u n j 1)] + µw 1 [g(u n j+1) 2g(u n j ) + g(u n j 1)] ū n+1 j has weak monotonicity, thus u n j 0 implies ū n+1 j 0 The same 1D limiter can be applied twice to for enforcing the bound 23 / 42

TVB limiter The result is still true if adding the TVB limiter in Bernardo Cockburn and Chi-Wang Shu Nonlinearly stable compact schemes for shock calculations SIAM Journal on Numerical Analysis, 31(3):607-627, 1994 Using both TVB limiter and the simple local bound-preserving limiter can eliminate overshoot/undershoot and oscillations without losing accuracy/conservation 24 / 42

2D Incompressible Flow The stream function formulation of 2D incompressible Navier-Stokes equations: ω t + (uω) x + (vω) y = ω u = ψ y, v = ψ x ψ = ω The weak monotonicity holds But to have H(M, M, M) = M, we need a discrete incompressible condition: [ 1 1 x 6 u i+1,j 1 + 4 6 u i+1,j + 1 6 u i+1,j+1 1 6 u i 1,j 1 4 6 u i 1,j 1 ] 6 u i 1,j+1 + 1 [ 1 y 6 v i+1,j+1 + 4 6 v i,j+1 + 1 6 v i 1,j+1 1 6 v i+1,j 1 4 6 v i,j 1 1 ] 6 v i 1,j 1 = 0 25 / 42

Inflow-outflow boundary conditions For the equation u t + f(u) x = 0 on [0, 1] with initial value u(x, 0) = u 0 (x) and inflow boundary condition u(0, t) = g(t) We have the 4th-order compact finite difference d 1 dt 6 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 u 0 u 1 u 3 u N u N+1 = 1 2 x 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 f 0 f 1 f 2 f N f N+1 26 / 42

Inflow-outflow boundary conditions With forward euler time discretization the scheme is ū n+1 i = ū n i t 2 x (f i+1 f i 1 ), 1 i N Since u 0 is known, to solve the previous linear system, we need to find a good approximation of u N+1 A cubic polynomial p j (x), obtained through enforcing its integral equals ū i over interval [x i 1, x i+1 ] for i = N 3, N 2, N 1, N We get A filter u N+1 = 2 3ūN 3 + 17 6 ūn 2 + 14 3 ūn 1 + 7 2ūN u N+1 = max(min(u N+1, ubound), lbound) should be used to make u N+1 [m, M] Next we can compute u i, 1 i N Apply the previous bound-preserving limiter to u i to make sure u i [m, M] Then we can update the value of ū i, 1 i N and compute the solution at next level 27 / 42

Dirichlet boundary conditions Consider u t + f(u) x = g(u) xx on [0, L] with initial value u(x, 0) = u 0 (x) and Dirichlet boundary condition u(0, t) = L(t), u(l, t) = R(t) Define W := W 1 W 2 = W 2 W 1, we will have and d dt (WU) + W 2D x f + W 2 F = W 1 D xx g + W 1 G (2) W 2 F + W 1 G = 1 18 u t,0+ 1 12 f x,0+ 5 12 x f 0+ 2 3 x 2 g 0 1 72 u t,0+ 1 24 x f 0+ 1 6 x 2 g 0 0 0 1 72 u t,n+1 1 24 x f N+1+ 1 6 x 2 g N+1 1 18 u t,n+1+ 1 12 f x,n+1 5 12 x f N+1+ 2 3 x 2 g N+1 (3) 28 / 42

Dirichlet boundary conditions We need to approximate f(u) x,0 and f(u) x,n+1 But to have weak monotonicity, only 3rd-order approximation can be made and f(u) x,0 = 1 x ( 11 6 f 0 + 3f 1 3 2 f 2 + 1 3 f 3) f(u) x,n+1 = 1 x ( 1 3 f N 2 + 3 2 f N 1 3f N + 11 6 f N+1) Then we plug these approximations in and rearrange (2) Finally we have the space discretization d dt Wũ = D x f + Dxx g 29 / 42

D x = 1 24 x W = 1 72 D xx = 1 6 x 2 24 246 5 5 38 5 42 5 84 5 6 5 1 14 42 14 1 78 5 1 14 42 14 1 6 5 2 5 84 5 246 5 1 10 0 10 1 24 5 42 5 24 5 1 10 0 10 1 2 5 78 42 38 5 5 5 12 5 6 5 1 2 6 2 1 1 2 6 2 1 6 5 12 5 42 5 24 5 N (N+2) N (N+2) N (N+2), ũ = u 0 u 1 u N u N+1 f 0 f 1, f =, g = f N f N+1 g 0 g 1 g N g N+1 (N+2) 1 (N+2) 1 (N+2) 1 Weak monotonicity and consistency hold The previous limiter can be applied to preserve the bound 2D case can be treated similarily,(4),(5) (6) 30 / 42

6th and 8th Order Scheme 6th order first derivative approximation with the weak monotonicity: βf i 2 + αf i 1 + f i + αf i+1 + βf i+2 = b f i+2 f i 2 4 x + a f i+1 f i 1 2, β = 1 12 ( 1 + 3α), a = 2 1 (8 3α), b = 9 18 ( 17 + 57α), α > 1 3 If we let α = 4 9, β = 1 36, a = 40 27, b = 25 54, we acctually have the 8th order with the weak monotonicity 6th-order second derivative approximation with the weak monotonicity: βf i 2 + αf i 1 + f i + αf i+1 + βf i+2 = b f i+2 2f i + f i 2 4 x 2 + a f i+1 2f i + f i 1 2, a = 78α + 48, b = 31 291α 36, α > 0 62 If we let α = 344 1179, we acctually have the 8th order scheme with the weak monotonicity 31 / 42

Non-uniform Grids Consider a mapping for the physical non-uniform grid points x j on the interval [a, b] to uniform reference grid points ξ j on the same interval [a, b] Let ξ(x) denote such a map and dξ(x) dx 0 Then a convection diffusion equation in the form of can be written as: tû + u t + f(u) x = g(u) xx ( ) dξ dx d2 ξ dx 2 ξ f(û) = ( ) dξ 2 2 dx ξ 2 g(û) Therefore, it is equivalent to solve the following equation u t + c(x)f(u) x = d(x)g(u) xx, d(x) > 0 32 / 42

Accuracy Test SSP 4th-order multistep for 2D linear convection: Mesh L error order L 1 error order 20 20 581E-5-368E-5-40 40 352E-6 405 225E-6 403 80 80 217E-7 402 138E-7 403 160 160 135E-8 401 857E-9 401 SSP RK4 for 1D linear convection: Mesh L error order L 1 error order 40 272E-5-167E-5-80 177E-6 395 108E-6 394 160 252E-7 281 837E-8 369 320 117E-7 111 120E-8 281 33 / 42

SSP 4th-order multistep for 1D convection-diffusion equation: Mesh L error order L 1 error order 40 155E-6-992E-7-80 972E-8 400 619E-8 400 160 607E-9 400 387E-9 400 320 376E-10 401 240E-10 401 34 / 42

1D burgers equation when the exact solution has a shock 2 2 Numerical Exact Numerical Exact 15 15 1 1 05 05 0 0-05 -05-1 -3-2 -1 0 1 2 3-3 -2-1 0 1 2 3 Left: numerical solutions without bound-preserving limiter Right: numerical solutions with bound-preserving limiter -1 35 / 42

12 4th order Compact FD with limiter Exact solution of u t =(u 8 ) xx 1 08 6 5 4 1 08 06 04 02 06 04 3 2 0-02 -04 02 0-6 -4-2 0 2 4 6 1 2 4 6-06 -08-1 Left: degenerate nonlinear parabolic equation u t = (u 8 ) xx, the numerical solution is strictly nonnegative Right: the vortex patch test for 2D incompressible NS equation with Reynolds number 3000, the solution is strictly bounded by 1 and 1 36 / 42

2D Incompressible flow with Reynolds number 3000: double shear layer 2 4 6 X 1 2 3 4 5 6 Y -4-2 0 2 4 The solution is strictly in the bound 37 / 42

8th order scheme for linear convection equation: Mesh L error order L 1 error order 20 314E-6-199E-6-40 713E-8 546 455E-8 545 80 271E-10 804 173E-10 804 160 143E-12 757 724E-12 790 8th order scheme for burgers equation: Mesh L error order L 1 error order 20 118E-3-337E-4-40 257E-5 552 223E-6 724 80 523E-8 894 297E-9 956 160 177E-10 821 990E-12 823 38 / 42

Inflow-outflow Boundary condition: 2 15 18 Numerical Exact 14 Numerical Exact 13 16 12 14 11 12 1 1 09 08 08 07 06 06 04 0 1 2 3 4 5 6 7 05 0 1 2 3 4 5 6 7 Burgers equation u t + ( 1 2 u2 ) xx = 0, when the solution is positive Left: numerical solution without bound-preserving limiter Right: numerical solution with bound-preserving limiter The solution is strictly bounded by 05 and 15 39 / 42

SSP 4th-order multistep for 1D convection diffusion equation with Dirichlet boundary condition: Mesh L error order L 1 error order 40 379E-5-203E-5-80 316E-6 358 116E-6 413 160 379E-7 306 126E-7 320 320 481E-8 298 186E-8 275 640 126E-9 525 472E-10 531 40 / 42

Generalization to Systems? Try to use the idea of FV? 41 / 42

The End 42 / 42