Jie Shen

Purdue University | Purdue · Department of Mathematics

We present a novel optimization algorithm, element-wise relaxed scalar auxiliary variable (E-RSAV), that satisfies an unconditional energy dissipation law and exhibits improved alignment between the modified and the original energy. Our algorithm features rigorous proofs of linear convergence in the convex setting. Furthermore, we present a simple...

For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained minimization in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization, which can be efficiently solved by first order optimiz...

Image inpainting models and the corresponding numerical algorithms play key roles in image processing. At present, the visual output of the oscillatory inpainting area is usually not natural. In this paper, we propose an image inpainting model based on the Ginzburg-Landau functional and \({H}^{-1}\)-norm. In the model, the \({H}^{-1}\)-fidelity ter...

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seed numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network...

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxil- iary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions [11, Huang and Shen, SIAM J. Numer. Anal., 2021] to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key i...

We extend the fictitious domain spectral method presented in Gu and Shen (SIAM J Sci Comput 43:A309–A329, 2021) for elliptic PDEs in bounded domains to the Helmhotlz equation in exterior domains. We first reduce the problem in an exterior domain to a bounded domain using the exact Dirichlet-to-Neumann operator. Next, we formulate the reduced proble...

We propose in this paper a new minimization algorithm based on a slightly modified version of the scalar auxiliary variable (SAV) approach coupled with a relaxation step and an adaptive strategy. It enjoys several distinct advantages over popular gradient based methods: (i) it is unconditionally energy diminishing with a modified energy which is in...

We carry out a rigorous error analysis of the first-order semi-discrete (in time) consistent splitting scheme coupled with a generalized scalar auxiliary variable (GSAV) approach for the Navier-Stokes equations with no-slip boundary conditions. The scheme is linear, unconditionally stable, and only requires solving a sequence of Poisson type equati...

The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme, which are important properties of the original system. In this paper, we establish the error estimates of this...

When a fluid-filled cube rotating rapidly about an axis passing through two opposite vertices is subjected to harmonic modulations of its rotation rate (librations) at a modulation frequency that is 2/ √ 3 times the mean rotation frequency, all walls of the cube have critical reflection slopes. As such, all inertial wave beams emitted from edges an...

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)$ with constant $\sigma\not=0$ and usual initial conditions at $t=-1,$ are associated with the gener...

We construct and analyze a numerical scheme based on the truly consistent splitting approach in time and the MAC discretization in space for the time dependent Stokes equations. The scheme only requires solving several Poisson type equations for the velocity and pressure at each time step. We establish the equivalence between two different formulat...

It has been a continuing challenge to carry out simulations at time and spatial scales compatible with practical experimental observations. Here we implement a novel scalar auxiliary variable (SAV) scheme introduced in (Shen et al., 2018) for phase-field equations to drastically improve the numerical accuracy, efficiency and stability. We first ben...

We develop in this paper two classes of length preserving schemes for the Landau-Lifshitz equation based on two different Lagrange multiplier approaches. In the first approach, the Lagrange multiplier $\lambda(\bx,t)$ equals to $|\nabla m(\bx,t)|^2$ at the continuous level, while in the second approach, the Lagrange multiplier $\lambda(\bx,t)$ is i...

The scalar auxiliary variable (SAV) approach [31] and its generalized version GSAV proposed in [20] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccura...

We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space–time Lagrange multiplier to enforce the positivity with the Karush–Kuhn–Tucker (KKT) conditions. We then use a predictor–corrector approach to construct a class of positivity schemes: with a gene...

We construct a new class of efficient implicit–explicit (IMEX) BDFk schemes combined with a scalar auxiliary variable (SAV) approach for general dissipative systems. We show that these schemes are unconditionally stable, and lead to a uniform bound of the numerical solution in the norm based on the principal linear operator in the free energy. Base...

We construct two classes of time discretization schemes for fourth order nonlinear equations by combing a function transform approach with the scalar auxiliary variable (SAV) approach. A suitable function transform ensures that the schemes are bound/positivity preserving, while the SAV approach enables us to construct unconditionally stable schemes...

A new class of time discretization schemes for the Navier-Stokes equations with non-periodic boundary conditions is constructed by combining the SAV approach for general dissipative systems in [15] and the consistent splitting schemes in [10]. The new schemes are unconditionally stable, only require solving linear equations with constant coefficien...

We construct first- and second-order time discretization schemes for the Cahn–Hilliard–Navier–Stokes system based on the multiple scalar auxiliary variables (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier–Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and only require...

The scalar auxiliary variable (SAV) approach \cite{shen2018scalar} and its generalized version GSAV proposed in \cite{huang2020highly} are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipati...

We construct three efficient and accurate numerical methods for solving the Klein–Gordon–Schrödinger (KGS) equations with/without damping terms. The first one is based on the original SAV approach, it preserves a modified Hamiltonian but does not preserve the wave energy. The second one is based on the Lagrange multiplier SAV approach, it preserves...

We develop a special phase field/diffusive interface method to model the nuclear architecture reorganization process. In particular, we use a Lagrange multiplier approach in the phase field model to preserve the specific physical and geometrical constraints for the biological events. We develop several efficient and robust linear and weakly nonline...

In this paper, we construct efficient schemes based on the scalar auxiliary variable (SAV) block-centered finite difference method for the modified phase field crystal (MPFC) equation, which is a sixth-order nonlinear damped wave equation. The schemes are linear, conserve mass and unconditionally dissipate a pseudo energy. We prove rigorously secon...

In the second part of this series, we use the Lagrange multiplier approach proposed in the first part \cite{CheS21} to construct efficient and accurate bound and/or mass preserving schemes for a class of semi-linear and quasi-linear parabolic equations. We establish stability results under a general setting, and carry out an error analysis for a se...

We consider a second-order SAV scheme for the nonlinear Schrödinger equation in the whole space with typical generalized nonlinearities, and carry out a rigorous error analysis. We also develop a fully discretized SAV scheme with Hermite–Galerkin approximation for the space variables, and present numerical experiments to validate our theoretical re...

We develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the non-local fractional Lapl...

We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach is based on expanding a generic spatial discretization, which is not necessarily positivity preserving, by introducing a space-time Lagrange multiplier coupled with Karush-Kuhn-Tucker (KKT) conditions to preserve p...

We develop a set of numerical schemes for the Poisson–Nernst–Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results ar...

Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this me...

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation o...

We construct and analyze first- and second-order implicit-explicit (IMEX) schemes based on the scalar auxiliary variable (SAV) approach for the magneto-hydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable...

In this paper, based on the Scalar Auxiliary Variable (SAV) approach [40, 41] and a newly proposed Lagrange multiplier (LagM) approach [21, 20] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of...

We construct new first- and second-order pressure correctionschemes using the scalar auxiliary variable approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error e...

We construct high-order semi-discrete-in-time and fully discrete (with Fourier-Galerkin in space) schemes for the incompressible Navier-Stokes equations with periodic boundary conditions, and carry out corresponding error analysis. The schemes are of implicit-explicit type based on a scalar auxiliary variable (SAV) approach. It is shown that numeri...

We develop a special phase field/diffusive interface method to model the nuclear architecture reorganization process. In particular, we use a Lagrange multiplier approach in the phase field model to preserve the specific physical and geometrical constraints for the biological events. We develop several efficient and robust linear and weakly nonline...

We construct efficient implicit-explicit BDF$k$ scalar auxiliary variable (SAV) schemes for general dissipative systems. We show that these schemes are unconditionally stable, and lead to a uniform bound of the numerical solution in the norm based on the principal linear operator in the energy. Based on this uniform bound, we carry out a rigorous e...

We propose in this paper three generalized auxiliary scalar variable (G-SAV) approaches for developing, efficient energy stable numerical schemes for gradient systems. The first two G-SAV approaches allow a range of functions in the definition of the SAV variable, furthermore, the second G-SAV approach only requires the total free energy to be boun...

For numerical schemes to the incompressible Navier-Stokes equations with variable density, it is a critical property to preserve the bounds of density. A bound-preserving high order accurate scheme can be constructed by using high order discontinuous Galerkin (DG) methods or finite volume methods with a bound-preserving limiter for the density evol...

We propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz–Petrich model. The function space, initial and boundary conditions are carefully chosen so that it fixes the relative orientation and displacement, and we follow a gradient flow to let the interface...

We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $\log $ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose...

We construct first- and second-order time discretization schemes for the Cahn-Hilliard-Navier-Stokes system based on the multiple scalar auxiliary variables approach (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier-Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and onl...

We develop several efficient numerical schemes which preserve exactly the global constraints for constrained gradient flows. Our schemes are based on the scalar auxiliary variable (SAV) approach combined with the Lagrangian multiplier approach. They are as efficient as the SAV schemes for unconstrained gradient flows, i.e., only require solving lin...

We propose a new Lagrange multiplier approach to design unconditional energy stable schemes for gradient flows. The new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed SAV approach (Shen, Xu, and Yang 2018), but enjoys two additional advantages: (i) schemes based on the new approac...

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally...

We develop a set of numerical schemes for the Poisson--Nernst--Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results...

We develop a new Lagrangian approach — flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager [27], [28]. Its main advantage, comparing with numerical methods in Eulerian coordinates, is t...

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we a...

We develop accurate and efficient spectral methods for elliptic PDEs in complex domains using a fictitious domain approach. Two types of Petrov–Galerkin formulations with special trial and test functions are constructed, one is suitable only for the Poisson equation but with a rigorous error analysis, the other works for general elliptic equations...

We carry out in this paper a rigorous error analysis for a finite element discretization of the scalar auxiliary variable (SAV) schemes. The finite-element method we study is a Galerkin method with standard Lagrange elements based on a mixed variational formulation. We derive optimal error estimates for both the first- and second-order SAV schemes...

In this paper, we construct efficient schemes based on the scalar auxiliary variable (SAV) block-centered finite difference method for the modified phase field crystal (MPFC) equation, which is a sixth-order nonlinear damped wave equation. The schemes are linear, conserve mass and unconditionally dissipate a pseudo energy. We prove rigorously secon...

Single-image super-resolution reconstruction aims to obtain a high-resolution image from a low-resolution image. Since the super-resolution problem is ill-posed, it is common to use a regularization technique. However, the choice of the fidelity and regularization terms is not obvious, and it plays a major role in the quality of the desired high re...

We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions and apply them to solve several typical fractional differential equ...

A class of bound preserving and energy dissipative schemes for the porous medium equation are constructed in this paper. The schemes are based on a positivity preserving approach for Wasserstein gradient flow and a perturbation technique, and are shown to be uniquely solvable, bound preserving, and in the first-order case, also energy dissipative....

We construct new first- and second-order pressure correction schemes using the scalar auxiliary variable (SAV) approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require a sequence of solving Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous...

The operator splitting method has shown to be an effective approach for solving the linear complementarity problem for pricing American options. It has been successfully applied to various Black–Scholes models, and it is implementation friendly because the differential equation and the complementarity conditions are decoupled and easily solved on i...

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the \(d+1\) dimensional problem to a s...

We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbou...

We develop several efficient numerical schemes which preserve exactly the global constraints for constrained gradient flows. Our schemes are based on the SAV approach combined with the Lagrangian multiplier approach. They are as efficient as the SAV schemes for unconstrained gradient flows, i.e., only require solving linear equations with constant...

The velocity correction method has shown to be an effective approach for solving incompressible Navier–Stokes equations. It does not require the initial pressure and the inf-sup condition may not be needed. However, stability and convergence analyses have not been established for the nonlinear case. The challenge arises from the splitting associate...

We propose a new Lagrange Multiplier approach to design unconditional energy stable schemes for gradient flows. The new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed SAV approach \cite{SAV01}, but enjoys two additional advantages: (i) schemes based on the new approach dissipate t...

We develop a new Lagrangian approach --- flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager \cite{onsager1931reciprocal,onsager1931reciprocal2}. Its main advantage, comparing with nume...

A space–time Petrov–Galerkin spectral method for time fractional diffusion equations is developed in this paper. The Petrov–Galerkin method is used to simplify the computation of stiffness matrix but leads to full non-symmetric mass matrix. However, the matrix decomposition method based on eigen-decomposition is numerically unstable for non-symmetr...

We develop an efficient numerical scheme for the 3D mean-field spherical dynamo equation. The scheme is based on a semi-implicit discretization in time and a spectral method in space based on the divergence-free spherical harmonic functions. A special semi-implicit approach is proposed such that at each time step one only needs to solve a linear sy...

Two classes of efficient and robust schemes are proposed for the general multi-symplectic Hamiltonian systems using the invariant energy quadratization (IEQ) approach. The schemes are linear, second-order accurate, local energy-preserving, and preserve the global energy. They are not restricted to specific forms of the nonlinear part of the state f...

An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell scheme (MAC) is constructed for the Navier-Stokes equations. A particular feature of the scheme is that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried out to show that both velocity...

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the...

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we al...

In this paper, we construct a positivity-preserving Gauge–Uzawa method for the semi-discrete-in-time scheme of incompressible viscous flows with variable density, and establish its stability and error estimates. We also construct and implement a fully discrete scheme with finite elements in space and derive its positivity-preserving and stability r...

Efficient and accurate numerical schemes, based on the scalar auxiliary variable (SAV) approach, are proposed to find the ground state solutions of one- and multi-component Bose-Einstein Condensates (BECs). Two types of SAV schemes are proposed: the first is based on the original SAV scheme for the imaginary time gradient flow of BECs which is accu...

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and error analysis. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish s...

We present essential properties of the generalized Jacobi functions (GJFs) and their application to construct efficient and accurate spectral methods for a class of fractional differential equations. In particular, it is shown that GJFs allow us to effortlessly compute the stiffness matrices and resolve the leading singular term for a general class...

This article describes fundamental approaches for the numerical handling of problems arising in fractional calculus. This includes, in particular, methods for approximately computing fractional integrals and fractional derivatives, where the emphasis is placed on Caputo operators, as well as solvers for the associated differential and integral equa...