Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.

Multi-domain Spectral Immersed Interface Method for Solving Elliptic Equation with a Global Description of Discontinuous Functions

This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interlace method（IIM） divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two that the domain decomposition technique can placed around the discontinuity. The present reached, in spite of the enlarged computational discontinuous problems are tested to verify the present method. The results show reduce the error of the spectral IIM, especially when more collocation points are method is t：avorable for the reason that the same level of the accuracy can be domain.

The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method(IIM)to simulate a two-fluid flow of two immiscible fluids with different density and viscosity.Due to the surface tension and the discontinuous fluid properties,the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids.The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface.We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in[Xu,DCDS,Supplement 2009,pp.838-845].We test our method on some canonical two-fluid flows.The results demonstrate that the method can handle large density and viscosity ratios,is second-order accurate in the infinity norm,and conserves mass inside a closed interface.

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant,closed tube,driven by an internal source and sink.Fluid is assumed to be highly viscous with its motion described by Stokes flow.Model geometry is assumed to be axisymmetric,and the governing equations are implemented in axisymmetric cylindrical coordinates,which capture 3D flow dynamics with only 2D computations.We solve the model equations using a hybrid approach:we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink,with each part handled separately by means of an appropriate method.Because the singularly-supported surface forcings yield an unsmooth solution,that part of the solution is computed using the immersed interface method.Jump conditions are derived for the axisymmetric cylindrical coordinates.The velocity due to the source and sink is calculated along the tubular surface using boundary integrals.Numerical results are presented that indicate second-order accuracy of the method.

High Order Scheme for Schrödinger Equation with Discontinuous Potential I: Immersed Interface Method

The immersed interface method is modified to compute Schrödinger equation with discontinuous potential.By building the jump conditions of the solution into the finite difference approximation near the interface,this method can give at least second order convergence rate for the numerical solution on uniform cartesian grids.The accuracy of this algorithm is tested via several numerical examples.

An Immersed Interface Method for the Simulation of Inextensible Interfaces in Viscous Fluids

In this paper,an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow.The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility,and the resulting augmented system is solved by the GMRES method.In this work,the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation.The forces at the interface are calculated from the configuration of the interface and the computed augmented variable,and then applied to the fluid through the related jump conditions.The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method.The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow.Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously.These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.

An Iterative Two-Fluid Pressure Solver Based on the Immersed Interface Method

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm.The iteration is constructed by introducing an unsteady term in the pressure Poisson equation.In each iteration step,a Helmholtz equation is solved on the Cartesian grid using FFT.The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across twofluid interfaces.This solver can also be used to solve Poisson equations on irregular domains.

The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.

A Simple 3D Immersed InterfaceMethod for Stokes Flow with Singular Forces on Staggered Grids

In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.

Immersed Interface CIP for One Dimensional Hyperbolic Equations

The immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients.The proposed method achieves the third order of accuracy in time and space in the vicinity of the interface where the coefficients have jump discontinuities,which is the same order of accuracy of the standard CIP scheme.Some numerical tests are given to verify the accuracy of the proposed method.

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant viscosity and singular force along an interface.We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface.Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface,we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity,respectively.The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs.These two augmented inputs provide the interface information,so we assume that a level set function is given and its zero level set indicates the position of the interface.The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity,while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation.We perform a series of numerical experiments to solve two-and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature.Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.

Wave Propagation Across Acoustic/Biot’s Media:A Finite-Difference Method

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency Biot’s equations(in the porous medium).Interface conditions are introduced to model various hydraulic contacts between the two media:open pores,sealed pores,and imperfect pores.Well-posedness of the initial-boundary value problem is proven.Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context:a fourth-order ADER scheme with Strang splitting for timemarching;a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory;and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution.Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions,demonstrating the accuracy of the approach.

A second-order numerical method for elliptic equations with singular sources using local flter

The presence of Dirac delta function in differential equation can lead to a discontinuity,which may degrade the accuracy of related numerical methods.To improve the accuracy,a secondorder numerical method for elliptic equations with singular sources is introduced by employing a local kernel flter.In this method,the discontinuous equation is convoluted with the kernel function to obtain a more regular one.Then the original equation is replaced by this fltered equation around the singular points,to obtain discrete numerical form.The unchanged equations at the other points are discretized by using a central difference scheme.1D and 2D examples are carried out to validate the correctness and accuracy of the present method.The results show that a second-order of accuracy can be obtained in the fltering framework with an appropriate integration rule.Furthermore,the present method does not need any jump condition,and also has extremely simple form that can be easily extended to high dimensional cases and complex geometry.

An Implementation ofMAC Grid-Based IIM-Stokes Solver for Incompressible Two-Phase Flows

Abstract.In this paper,a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces.The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method.The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity,and are interpolated using cubic splines and are then applied to the fluid through the jump conditions.The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-typemethod.The numerical results show that the overall scheme is second order accurate.The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions.The proposed method avoids solution of the pressure Poisson equation,and comparisons are made to show the advantages of time savings by the present method.The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.

The Sensitivity Analysis for the Flow Past Obstacles Problem with Respect to the Reynolds Number

In this paper,numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented.To carry out such analysis,at each time step,we need to solve the incompressible Navier-Stokes equations on irregular domains twice,one for the primary variables;the other is for the sensitivity variables with homogeneous boundary conditions.The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains.One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle.Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.

Numerical Study of Surfactant-Laden Drop-Drop Interactions

In this paper,we numerically investigate the effects of surfactant on dropdrop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in(Xu et al.,J.Comput.Phys.,212(2006),590–616).We find that surfactant plays a critical and nontrivial role in drop-drop interactions.In particular,we find that the minimum distance between the drops is a non-monotone function of the surfactant coverage and Capillary number.This non-monotonic behavior,which does not occur for clean drops,is found to be due to the presence of Marangoni forces along the drop interfaces.This suggests that there are non-monotonic conditions for coalescence of surfactant-laden drops,as observed in recent experiments of Leal and co-workers.Although our study is two-dimensional,we believe that drop-drop interactions in three-dimensional flows should be qualitatively similar as the Maragoni forces in the near contact region in 3D should have a similar effect.

An Augmented Approach for the Pressure Boundary Condition in a Stokes Flow

An augmented method is proposed for solving stationary incompressible Stokes equations with a Dirichlet boundary condition along parts of the boundary.In this approach,the normal derivative of the pressure along the parts of the boundary is introduced as an additional variable and it is solved by the GMRES iterative method.The dimension of the augmented variable in discretization is the number of grid points along the boundary which is O(N).Each GMRES iteration(or one matrix-vector multiplication)requires three fast Poisson solvers for the pressure and the velocity.In our numerical experiments,only a few iterations are needed.We have also combined the augmented approach for Stokes equations involving interfaces,discontinuities,and singularities.

Analysis and Computation for a Fluid Mixture Model

A fluid mixture model of tissue deformations has been studied in this paper.The model is a mixed system of nonlinear hyperbolic and elliptic partial differential equations.Both theoretical linear stability and numerical analysis are presented.Comparisons between standard numerical methods that utilize Runge-Kutta methods coupled with the WENO scheme and the immersed interface methods are given.Numerical examples are also presented.

A Fast High Order Iterative Solver for the Electromagnetic Scattering by Open Cavities Filled with the Inhomogeneous Media

The scattering of the open cavity filled with the inhomogeneous media is studied.The problem is discretized with a fourth order finite difference scheme and the immersed interfacemethod,resulting in a linear system of equations with the high order accurate solutions in the whole computational domain.To solve the system of equations,we design an efficient iterative solver,which is based on the fast Fourier transformation,and provides an ideal preconditioner for Krylov subspace method.Numerical experiments demonstrate the capability of the proposed fast high order iterative solver.