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Structure preserving numerical schemes for complex dissipative/conservative nonlinear systems

Jie Shen

Purdue University Collaborators: Jie Xu, Jiang Yang, Qing Cheng

ICERM Workshop on Numerical Methods and New Perspectives for Extended Liquid Crystalline Systems, Dec 9-13, 2019

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Outline

Motivation The scalar auxiliary variable (SAV) approach Application to a Q-tensor model SAV approach for Navier-Stokes equations and more general dissipative/conservative system The SAV approach with global constraints Concluding remarks

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Motivation

Many physical problems can be modeled by PDEs that take the form of gradient flows or Hamiltonian systems. Examples include Allen-Cahn, Cahn-Hilliard, phase-field models, liquid crystals, super conductivity, image processing, ...; nonlinear Schr¨

• dinger equation, Sine-Gordon equation, ...

Gradient flows or Hamiltonian systems are dynamically driven by a free energy or Hamiltonian E(φ), and take the form: ∂φ ∂t = −G δE(φ) δφ , where G is a positive operator (gradient flows) or a skew-symmetric operator (Hamiltonian systems), and satisfy a dissipative or conservative energy law: d dt E(φ) = −(G δE(φ) δφ , δE(φ) δφ ).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Some examples

heat equation: E(φ) =

• Ω

1 2|∇φ|2 and G = I;

Allen-Cahn: E(φ) =

• Ω( 1

2|∇φ|2 + 1 4ǫ2 (φ2 − 1)2) and G = I;

Cahn-Hilliard: E(φ) =

• Ω( ǫ

2|∇φ|2 + 1 4ǫ(φ2 − 1)2) and

G = −∆; Phase-field crystal: E(φ) =

• Ω( 1

4φ4 + α 2 φ2 − |∇φ|2 + 1 2|∆φ|2)

and G = −∆; L1 minimization: E(φ) =

• Ω |∇φ| and G = I;

Nonlinear Schr¨

• dinger equation:

E(φ) =

• Ω( 1

2|∇φ|2 + 1 2F(|φ|2)) and G = i;

KDV equation: E(φ) =

• Ω( 1

2|∂xφ|2 + φ3), G = ∂x.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Existing approaches for time discretization

Full implicit schemes: Can be unconditionally energy stable, but have to solve nonlinear equations at each step and may need ∆t sufficiently small to have a unique solution. stabilized linearly implicit schemes and ETD schemes: With some conditions, first-order schemes can be unconditionally energy stable; efficient but difficult to get to second- or higher-order. Convex splitting schemes: First-order schemes always unconditionally energy stable; difficult to get to second- or higher-order; still need to solve (much easier) nonlinear equations at each step. (invariant) energy quadratization (IEQ/EQ) schemes: Linear, second-order, unconditionally energy stable; work with a large class of gradient flows; need to solve coupled systems with variable coefficients.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

The scalar auxiliary variable (SAV) approach

Let E(φ) =

• Ω[ 1

2φ Lφ + F(φ)]dx and assume that

E1(φ) :=

• Ω F(φ)dx is bounded from below, i.e., E1(φ) > −C0 for

some C0 > 0. We introduce one scalar auxiliary variable (SAV): r(t) =

• E1(φ) + C0 so that E(φ) =
• Ω

1 2φ Lφdx + r2(t) − C0. Then, the original system ∂φ

∂t = −G δE(φ) δφ

can be recast as: ∂φ ∂t = −Gµ µ = Lφ + r(t)

• E1(φ) + C0

F ′(φ) rt = 1 2

• E1(φ) + C0
• Ω

F ′(φ)φtdx.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

The SAV scheme based on Crank-Nicolson

Second-order CN-AB scheme: φn+1 − φn ∆t = −Gµn+1/2, µn+1/2 = Lφn+1/2 + rn+1/2

• E1[˜

φn+1/2] + C0 F ′(˜ φn+1/2), rn+1 − rn ∆t =

• Ω

F ′(˜ φn+1/2) 2

• E1[˜

φn+1/2] + C0 φn+1 − φn ∆t dx, where ψn+1/2 := ψn+1+ψn

2

and g(˜ φn+1/2) := 3g(φn)−g(φn−1)

2

. Taking the inner products of the above with µn+1/2, − φn+1−φn

∆t

and 2rn+1/2, respectively, one derives the following energy law: 1 2(φn+1, Lφn+1)+|rn+1|2−1 2(φn, Lφn)−|rn|2 = −(Gµn+1/2, µn+1/2), i.e., it maintains the dissipation rate for dissipative systems and is energy conserving for conservative systems.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Difference with a semi-implicit scheme

Eliminating rn+1 from the above scheme, we find φn+1 − φn ∆t = −Gµn+1/2, µn+1/2 = Lφn+1/2 + rn

• E1[˜

φn+1/2] + C0 F ′(˜ φn+1/2) +

• Ω

F ′(˜ φn+1/2)(φn+1 − φn) dx F ′(˜ φn+1/2) 4(E1[˜ φn+1/2] + C0) . Without the last term, the scheme is a semi-implicit scheme. The last term serves as an extra stabilizing term!

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Efficient implementation

We can write the schemes as a matrix system   c1I G −L c2I ∗ ∗ c3     φn+1 µn+1 rn+1   = ¯ bn, So we can solve rn+1 with a block Gaussian elimination, which requires solving a system with constant coefficients of the form c1I G −L c2I φ µ

• = ¯

b. With rn+1 known, we can obtain (φn+1, µn+1) by solving one more equation in the above form. So the cost is essentially twice the cost of a semi-implicit scheme, but it enjoys many additional benefits.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Convergence and error analysis

The SAV schemes are semi-implicit schemes. Previous stability and error analysis on semi-implicit schemes usually assume the uniform Lipschitz condition on the second derivative of the nonlinear free energy, which is not satisfied by even the double-well potential. Thanks to the unconditional energy stability of the SAV schemes, we can derive H2 bounds for the numerical solution under mild conditions on the free energy. The H2 bounds on the numerical solution will enable us to establish the convergence, and with additional smoothness assumption, the error estimates. Error estimates for the SAV approach have been established for (i) semi-discrete case (S. & J. Xu, SINUM ’18); (ii) fully discrete with finite-elements (H. Chen & S.); (iii) fully discrete with finite-differences (X. Li., & H. Rui & S., Math Comp).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

SAV approach with stabilization

If the nonlinear term is too ”strong”, the SAV approach may require restrictive time steps for accuracy. However, this situation can be easily improved with a stabilization. Given ǫ ≪ 1. Consider the free energy E(φ) = 1 2|∇φ|2 + 1 ǫ2 F(φ). Then, the SAV approach with E1(φ) =

• 1

ǫ2 F(φ) will require

small time steps to get accurate results. Choose S > 0, and split the free energy as follows: E(φ) =

• (1

2|∇φ|2 + S ǫ2 φ2) + 1 ǫ2 (F(φ) − Sφ2). We still have

• 1

ǫ2 (F(φ) − Sφ2) > −C0, so SAV can be

applied with this splitting, and leads to much improved results.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

1 2 3 4 5 6

• 1.5
• 1
• 0.5

0.5 1 1.5

small time step SAVnoSTA SAVwithSTA

1 2 3 4 5 6

• 1.5
• 1
• 0.5

0.5 1 1.5

large time step SAVnoSTA SAVwithSTA

Figure: (Effect of stabilization) The solution at T = 0.1. Left: ∆t = 10−4; Right: ∆t = 4 × 10−3. The red dashed lines represent solutions with stabilization, while the black solid lines represent solutions without stabilization.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Adaptive time stepping

Thanks to its unconditionally energy stability, one can (and should) couple the scheme with an adaptive time stepping strategy. Note that |rn+1 − 1| serves as a simple and effective criterion.

Figure: Numerical comparisons among small time steps, adaptive time steps, and large time steps

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

SAV approach for the Q-tensor model

Consider the Landau-de Gennes free energy written as E[Q] = Eb + Ee with Eb(Q) =

• Ω

fb(Q)dx =

• Ω

[a 2trQ2 − b 3trQ3 + c 4(trQ2)2]dx, Ee(Q) =

• Ω

[L1 2 |∇Q|2 + L2 2

3

• k=1

∂iQik∂jQjk + L3 2

3

• k=1

∂iQjk∂jQik]dx, where L1, L1 + L2 + L3 > 0 so that Eb ≥ 0, and c > 0 such that Eb(Q) + C0 > 0 for some C0 > 0.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

The SAV formulation

Let LQ = δEe

δQ , and introduce a SAV

r(t) =

• Eb + C0.

We can rewrite the system as: ∂Q ∂t = −µ, µ = LQ + r(t) √Eb + C0 δEb δQ ; dr dt = 1 2√Eb + C0 (δEb δQ , ∂Q ∂t ).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Then, the SAV/CN scheme is: Qn+1 − Qn ∆t = − µn+1/2, µn+1/2 =LQn+1 + Qn 2 + rn+1 + rn 2

• Eb[ ¯

Qn+1/2] + C0 δEb δQ [ ¯ Qn+1/2], rn+1 − rn = 1 2

• Eb[ ¯

Qn+1/2] + C0 δEb δQ [ ¯ Qn+1/2], Qn+1 − Qn . One can easily show that the above scheme is unconditionally energy stable, and at each time step, one only needs to solve two equations of the form: (I + λL)Q = g. With the periodic boundary condition, the Fourier spectral method applied to the above equation leads to a 5 × 5 linear system for each of the Fourier mode. Hence it is extremely efficient and accurate.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

The Navier-Stokes equations

Consider the NSEs in a bounded domain Ω: ut + (u · ∇)u = ν∆u − ∇p, u|∂Ω = 0; and ∇ · u = 0. The NSE is not a gradient flow but it satisfy an energy dissipation law: 1 2 d dt

• |u|2 = −ν
• Ω

|∇u|2.

• Q. Can we construct an unconditionally energy stable scheme with

explicit treatment of nonlinear term for NSE?

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

A nonlinear SAV formulation (Lin, S. Dong & Z. Yang ’18-’19)

Let E(u) =

• Ω

1 2|u|2dx + δ and R(t) =

• E(u(t)). We rewrite

NSE as ut + R(t)

• E(u(t))

(u · ∇)u = ν∆u − ∇p, u|∂Ω = 0; ∇ · u = 0; 2R(t)R′(t) = (ut, u) = (ut + R(t)

• E(u(t))

(u · ∇)u, u). With R(0) =

• 1

2 d dt

• |u(·, 0)|2 + δ, the above system is

equivalent to the original NSE.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

SAV approach with pressure-correction

˜ un+1 − un ∆t + Rn+1

• E(un+1)

(un · ∇)un = ν∆˜ un+1 − ∇pn, ˜ un+1|∂Ω = 0; un+1 − ˜ un+1 ∆t + ∇(pn+1 − pn) = 0; ∇ · un+1 = 0, un+1 · n|∂Ω = 0; 2Rn+1 Rn+1 − Rn ∆t = (un+1 − un ∆t + Rn+1

• E(un+1)

(un · ∇)un, un+1). We can easily show that |Rn+1|2+1 2(∆t)2∇pn+12−|Rn|2−1 2(∆t)2∇pn2 ≤ −ν∆t∇un+12. Second-order scheme can also be constructed.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

How to solve the coupled system?

Let us denote Sn+1 =

Rn+1

√

E(un+1) and set

˜ un+1 = ˜ un+1

1

+ Sn+1 ˜ un+1

2

, un+1 = un+1

1

+ Sn+1un+1

2

, pn+1 = pn+1

1

+ Sn+1pn+1

2

. One can check that ˜ un+1

i

(i = 1, 2) satisfy: ˜ un+1

1

− un ∆t = ν∆˜ un+1

1

− ∇pn

1,

˜ un+1

1

|∂Ω = 0; ˜ un+1

2

∆t + (un · ∇)un = ν∆˜ un+1

2

− ∇pn

2,

˜ un+1

2

|∂Ω = 0; and that un+1

i

, pn+1

i

(i = 1, 2) satisfy: un+1

i

− ˜ un+1

i

∆t + ∇(pn+1

i

− pn

i ) = 0;

∇ · un+1

i

= 0, un+1

i

· n|∂Ω = 0.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Once ˜ un+1

i

, un+1

i

, pn+1

i

(i = 1, 2) are known, we can solve Sn+1 by solving a nonlinear (algebraic) quadratic equation. Remarks: The scheme is very efficient: at each time step, it requires

• nly solving two sets of Poisson type equations.

S − 1 provides a natural estimator for adaptive time stepping. Ample numerical results by S. Dong et al. show that the SAV approach is more efficient and robust than the usual semi-implicit schemes. This approach, coupled with the SAV approach for Q-tensor model, can be used to construct very efficient SAV schemes for hydrodynamic Q-tensor model for liquid crystal flows (cf. related work by Qi Wang, Jia Zhao and Xiaofeng Yang).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

SAV approach for general dissipative/conservative systems (Yang & Dong ’19; Huang, S. & Yang ’19)

Consider a general dissipative system φt + Aφ + F(φ) = 0, where A is a linear positive operator and F(φ) is a nonlinear

• perator, with an energy dissipation/conservative law

d dt E(φ) = −(Gφ, φ), with E(u) is a certain energy bounded from below, G being a positive (for dissipative) or skew-symmetric (for conservative)

• perator.

Introducing a SAV, r(t) = E(φ(t)) + C0 > 0, we can rewrite the system as φt + Aφ + r(t) E(φ(t)) + C0 F(φ) = 0, rt = −(Gφ, φ).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

A linear, unconditionally energy stable SAV scheme

φn+1 − φn δt + Aφn+1 + ξn+1F(φn) = 0, rn+1 − rn δt = −(G ¯ φn+1, ¯ φn+1)ξn+1, where ξn+1 =

rn+1 E(¯ φn+1)+C0 , and ¯

φn+1 is defined as follows: Setting φn+1 = φn+1

1

+ ξn+1φn+1

2

, we find from the above that φn+1

1

− φn δt + Aφn+1

1

= 0, φn+1

2

δt + Aφn+1

2

+ F(φn) = 0. Then, we set ¯ φn+1 = φn+1

1

+ φn+1

2

.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Once we find φn+1

i

(i = 1, 2) and ¯ φn+1, we can obtain rn+1 directly from the second equation by rn+1 = rn/(1 + δt E(¯ φn+1) + C0 (G ¯ φn+1, ¯ φn+1)). We observe that rn+1 ≥ 0 if rn ≥ 0. Hence, the scheme is unconditionally energy stable! It only requires solving two linear systems with constant coefficients, so it is very efficient. |ξn+1 − 1| is a natural estimator for adaptive time stepping. It can be coupled with any high-order time discretization and is still unconditionally energy stable.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Gradient flows with global constraints

Consider, e.g, the phase-field vesicle membrane model with the bending energy (Du, Liu & Wang ’04): Eb(φ) = ǫ 2

• Ω
• − ∆φ + 1

ǫ2 G(φ) 2 dx = ǫ 2

• Ω
• (−∆φ)2 + 6

ǫ2 φ2|∇φ|2 + 1 ǫ4 (G(φ))2 dx, (where F(φ) = (1 − φ2)2 and G(φ) = F ′(φ)) with constraints that the volume and surface area of the vesicles A(φ) = 1 2

• Ω

(φ + 1)dx and B(φ) =

• Ω

ǫ 2|∇φ|2 + 1 ǫ F(φ)

• dx

are conserved.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Penalty approach with MSAV (Q. Cheng & S., SISC ’18)

Consider the gradient flow with the penalized total energy: Etot(φ) = Eb(φ) + 1 2γ

• A(φ) − α

2 + 1 2η

• B(φ) − β

2 , where γ and η are two small parameters, and α, β represent the initial volume and surface area. One can then apply directly the SAV approaches with a single SAV, but very small time steps are needed to obtain accurate numerical results due to multiple small parameters in the penalized free energy. Using multiple SAV approaches can greatly increase the allowable time steps, but very small time steps are still required if one wants the constraints to be satisfied with very high accuracy.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

δt allowed MSAV: ǫ MSAV: η 4 × 10−4 10−5 10−5 1 × 10−4 10−7 10−7 2 × 10−5 10−9 10−9 2 × 10−6 10−11 10−11 1 × 10−6 10−12 10−12

Table: Largest time step allowed for MSAV scheme with various Penalty parameters ǫ and η

An alternative approach is to introduce Lagrange multipliers to enforce exactly the constraints.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

The Lagrange multiplier approach

Consider again a system with free energy E(φ) =

• Ω

1 2Lφ · φ + F(φ)dx, under a global constraint d dt H(φ) = 0 with H(φ) =

• Ω

h(φ)dx. Introducing a Lagrange multiplier λ(t), the general gradient flow with the above free energy under the constraint takes the following form φt = −Gµ, µ = Lφ + F ′(φ) − λ(t)δH δφ , d dt H(φ) = 0,

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

A SAV formulation

Introduce a SAV, r(t) =

• Ω F(φ)dx + C0, we obtain an

equivalent system ∂tφ = −Gµ, µ = Lφ + r(t)

• Ω F(φ)dx + C0

F ′(φ) − λ(t)δH δφ , d dt H(φ) = 0, rt = 1 2

• Ω F(φ)dx + C0

(F ′(φ), φt). One can then construct an efficient SAV scheme which enforces the constraint exactly, but unfortunately, one can not prove that it is unconditionally energy stable.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

A Lagrange multiplier formulation

Instead of introducing the SAV r(t), we introduce another Lagrange multiplier to enforce the energy dissipation: ∂tφ = −Gµ, µ = Lφ + η(t)F ′(φ) − λ(t)δH δφ , d dt H(φ) = 0, d dt

• Ω

F(φ)dx = η(t)(F ′(φ), φt) − λ(t)(δH δφ , φt).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Then, a second-order scheme is as follows: φn+1 − φn δt = −Gµn+1/2, µn+1/2 = Lφn+1/2 + ηn+1/2F ′(φ∗,n+1/2) − λn+1/2(δH δφ )∗,n+1/2, H(φn+1) = H(φ0),

• Ω

F(φn+1) − F(φn)dx = ηn+1/2(F ′(φ∗,n+1/2), φn+1 − φn) −λn+1/2((δH δφ )∗,n+1/2, φn+1 − φn), where f n+1/2 = 1

2(f n+1 + f n) and f ∗,n+1/2 = 1 2(3f n − f n−1) for

any sequence {f n}.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

We have the following discrete energy law: Theorem The scheme is unconditionally energy stable in the sense that E(φn+1) − E(φn) = −δt(Gµn+1/2, µn+1/2), where E(φ) is the original energy. Unlike the original SAV approach, the scheme is (mildly) nonlinear. But it can be solved as efficiently as the original SAV approach.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Indeed, writing φn+1 = φn+1

1

+ ηn+1/2φn+1

2

+ λn+1/2φn+1

3

, µn+1 = µn+1

1

+ ηn+1/2µn+1

2

+ λn+1/2µn+1

3

, we find that (φn+1

i

, µn+1

i

) (i = 1, 2, 3) can be determined as follows: φn+1

1

− φn δt = −Gµn+1/2

1

, µn+1/2

1

= Lφn+1/2

1

; φn+1

2

δt = −Gµn+1/2

2

, µn+1/2

2

= Lφn+1/2

2

+ F ′(φ∗,n+1/2); φn+1

3

δt = −Gµn+1/2

3

, µn+1/2

3

= Lφn+1/2

3

− (δH δφ )∗,n+1/2.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Once we determine (φn+1

i

, µn+1

i

) (i = 1, 2, 3) from the above, we can plug them into the scheme to obtain a nonlinear ALGEBRAIC system for (ηn+1/2, λn+1/2). The coupled nonlinear algebraic system for (ηn+1/2, λn+1/2) can be solved by Newton iteration. Since the exact solution η(t) ≡ 1, we can use 1 as the initial guess for ηn+1/2, and use a linear scheme to produce an initial guess for λn+1/2. With the above initial guess, the Newton iteration will converge in just a few steps with negligible cost compared to the total cost.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Figure: Evolution of the volume difference and surface area by the MSAV scheme (ǫ = 10−9) and the Lagrange multiplier scheme.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Figure: Collision of four spherical vesicles with the volume and surface area constraints (i.e., η = γ = 0.001). Snapshots of the iso-surfaces of φ = 0 at t = 0, 0.005,0.002, 0.1, 0.5,2.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

An optimal partition problem with multiple constraints

E(φ) =

• Ω

(1 2|∇φ|2 + F(φ))dx, with F(φ) = 1

ǫ2 m

• i=1
• j<i

φ2

i φ2 j satisfying the norm constraints

Hj(φ) :=

• Ω

|φj|2dx = 1, j = 1, 2, . . . , m. The corresponding gradient flow reads ∂tφj = −µj, µj = −∆φj − λj(t)φj + δF δφj , d dt

• Ω

|φj(x, t)|2dx = 0, j = 1, 2, . . . , m, with initial condition

• Ω |φj(x, 0)|2dx = 1.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Figure: A 10-subdomain partition: initial partition and subdomains at times t = 0, 0.05, 0.5, 2 computed by the Lagrangian multiplier approach.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Concluding remarks

The SAV approach enjoys the following advantages:

It is second-order unconditionally energy stable and can be extended to higher-order. It leads to linear, decoupled equations with CONSTANT coefficients, even for gradient flows with multiple components. It can be applied to a larger class of gradient flows/Hamiltonian systems, and can be combined with any consistent Galerkin type spatial discretization. Rigorous convergence and error analyses are established without the usually assumed uniform Lipschitz condition.

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

Some other new developments related to SAV

Effective SAV schemes for computing ground states of single and multi component BECs (S. & Zhuang ’19); New SAV schemes which only require solving one linear system per time step and high-order time adaptivity (Huang, S., Yang ’19); Stabilized SAVs: for problems with high-order nonlinear terms,

• ne may introduce additional stabilizing terms (X. Yang ’18);

SAV approach for more general dissipative systems : Z. Yang& S. Dong ’19, Huang, S. Yang ’19; SAV approach applied to linearly implicit high-order R-K schemes: G. Akrivis, B. Li & D. Li. (SISC ’19).

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative

References: ”A new class of efficient and robust energy stable schemes for gradient flows”, by J. Shen, Jie Xu and Jiang Yang, SIAM Review 61:474-506, 2019. ”Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows”, by Jie Shen and Jie Xu, SIAM J. Numer. Anal., 2018. ”A highly efficient and accurate new SAV approach for gradient flows”, by F. Huang, J. Shen & Z. Yang (submitted). ”Unconditionally positivity preserving and energy dissipative schemes for Poisson–Nernst–Planck equations”, by Jie Shen and Jie Xu, submitted.

Thank you!

Jie Shen Structure preserving numerical schemes for complex dissipative/conservative