Structure-preserving Scheme Research Articles

Novel conformal structure-preserving schemes for the linearly damped nonlinear Schrödinger equation

In this work, by utilizing the Strang splitting technique, some advanced conformal structure-preserving schemes are developed for solving the damped nonlinear Schrödinger equation (DNLSE). The proposed innovative numerical approaches, namely the high-order compact conformal multi-symplectic method and the conformal momentum-preserving method, have two key computational advantages. Firstly, these two approaches excel in conserving local structures, and more importantly, they are capable of maintaining exact energy dissipation rates in any time-space region, particularly under periodic boundary conditions. To validate the theoretical properties and demonstrate the efficacy and stability of these approaches in long-time integrations, we conduct through extensive numerical simulations involving both bright and dark solitons.

A structure-preserving algorithm for Vlasov–Darwin system conserving charge, canonical momentum, and Hamiltonian

In this paper, a conservative numerical method for the Vlasov–Darwin system is proposed. The Darwin model was assumed to be valid only in the Coulomb gauge, but recently this model has also been extended to the Lorenz gauge [1]. The Darwin model, based on the Lorenz gauge, exhibits a good symmetry between scalar and vector potentials, making the proofs of physical constraints relatively easy. In addition, the total energy was believed to be one of the conservative quantities of the Vlasov–Darwin system. However, the improved theory suggests that the Hamiltonian is the conservative quantity rather than the total energy. The structure-preserving scheme proposed in this paper exactly maintains the Lorenz gauge and the conservation laws of charge, canonical momentum, and Hamiltonian in discrete form.

A Poisson-bracket scheme for nonlinear shallow-water sloshing in an oscillating tank with irregular bottom surface

The mass-, energy-, and potential-enstrophy-conserving Poisson bracket numerical scheme introduced by Arakawa & Lamb (1981), and extended by Salmon (2004) and Stewart & Dellar (2016), is adapted to the problem of nonlinear shallow-water sloshing over a variable bottom surface in a rigid rectangular basin undergoing a prescribed coupled surge-sway motion. Adaptation to a finite domain requires a new approach to the boundary conditions at solid boundaries in the context of the Arakawa C grid. In this paper, the boundary condition at the vertical walls is taken to be vanishing of the normal derivative of the tangential velocity. This gives zero potential vorticity at the boundary, and is consistent with the material conservation of the potential vorticity. This condition, coupled to symmetric boundary conditions for the wave height arising from a reduced version of the evolution equations at boundaries, leads to an extension of the class of staggered C-grid Poisson-bracket schemes to interior flows with solid boundaries. The scheme is implemented, shown to preserve Casimirs, and applied to a range of problems in shallow water hydrodynamics including interaction of travelling hydraulic jumps at resonance, and X-type soliton interactions over a variable bottom surface. This structure-preserving scheme provides a robust building block for long-time simulation of floating ocean wave energy extractors.

An energy-stable variable-step L1 scheme for time-fractional Navier–Stokes equations

We propose a structure-preserving scheme and its error analysis for time-fractional Navier–Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when α→1. Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis of structure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.

Energy stable and structure-preserving schemes for the stochastic Galerkin shallow water equations

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

Novel structure-preserving schemes for stochastic Klein–Gordon–Schrödinger equations with additive noise

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings.

Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

A decoupled, linearly implicit and high-order structure-preserving scheme for Euler–Poincaré equations

It is challenging to develop high-order structure-preserving finite difference schemes for the modified two-component Euler–Poincaré equations due to the nonlinear terms and high-order derivative terms. To overcome the difficulties, we introduce a bi-variate function and carefully choose the intermediate average variable in the temporal discretization. Then, we obtain a decoupled and linearly implicit scheme. It is shown that the fully-discrete scheme can keep both the discrete mass and energy conserved. And the fully-discrete scheme has fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction. Several numerical examples are given to confirm the theoretical results.

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O(τ), where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme.

Pointwise second order convergence of structure-preserving scheme for the triple-coupled nonlinear Schrödinger equations

In this paper, we present a finite difference scheme for the triple-coupled Schrödinger equations (T-CNLS) in optics. The T-CNLS is approximated by Crank-Nicolson scheme in time and finite difference method in space. Some mathematical characters are investigated, such as structure-preserving properties, unique solvability, convergence in L∞ norm. Some numerical examples are reported to illustrate the theoretical results.

A new open-source library based on novel high-resolution structure-preserving convection schemes

This work develops a new finite volume convection scheme and open-source library for convection-dominated problems on unstructured grids. The formulated reconstruction operators can realize high-resolution and structure-preserving properties by adjusting numerical dissipation and anti-dissipation errors in a unified normalized-variable diagram. Then, this work develops a new open-source library based on the devised schemes within one of the most widely used general-purpose open-source software whose solution quality is intrinsically limited by the low accuracy of the conventional total variation diminishing schemes. The performance of the new library is first verified by some fundamental problems including pure advection and inviscid Euler equations. The numerical results of the accuracy tests show that the developed interpolation library can significantly reduce numerical errors with a minor increased computational cost compared with the conventional second-order schemes. The complex profile advection test demonstrates that the new library can offer an improved structure-preserving property that gives essentially oscillation-free solutions and preserves the structures of the passive transported scalar. Furthermore, the robustness of the new library is validated by compressible high-speed flow simulations with irregular geometries. Thus, this work presents a new library of high-resolution structure-preserving schemes for the accurate and efficient discretization of the convection term which plays an essential role in a wide range of computational fluid dynamics. The source code of the new scheme and library are released via DOI:10.5281/zenodo.8399000.

Second-Order Accurate Structure-Preserving Scheme for Solute Transport on Polygonal Meshes

Second-Order Accurate Structure-Preserving Scheme for Solute Transport on Polygonal Meshes

Compact Structure-Preserving Signatures with Almost Tight Security

In structure-preserving cryptography, every building block shares the same bilinear groups. These groups must be generated for a specific, a priori fixed security level, and thus, it is vital that the security reduction in all involved building blocks is as tight as possible. In this work, we present the first generic construction of structure-preserving signature schemes whose reduction cost is independent of the number of signing queries. Its chosen-message security is almost tightly reduced to the chosen-plaintext security of a structure-preserving public-key encryption scheme and the security of Groth–Sahai proof system. Technically, we adapt the adaptive partitioning technique by Hofheinz (Eurocrypt 2017) to the setting of structure-preserving signature schemes. To achieve a structure-preserving scheme, our new variant of the adaptive partitioning technique relies only on generic group operations in the scheme itself. Interestingly, however, we will use non-generic operations during our security analysis. Instantiated over asymmetric bilinear groups, the security of our concrete scheme is reduced to the external Diffie–Hellman assumption with linear reduction cost in the security parameter, independently of the number of signing queries. The signatures in our schemes consist of a larger number of group elements than those in other non-tight schemes, but can be verified faster, assuming their security reduction loss is compensated by increasing the security parameter to the next standard level.

Efficient energy structure-preserving schemes for three-dimensional Maxwell's equations

Two energy structure-preserving schemes are proposed for Maxwell's equations in three dimensions. The Maxwell's equations are split into several local one-dimensional subproblems which successfully reduces the scale of algebraic equations to be solved. To improve the convergence rate in space and to keep the sparsity of the resulting algebraic equations, the spatial derivatives are approximated by high order compact method. Some key indicators, such as stability, energy structure-preserving and convergence of the schemes are investigated. To make the theoretical more persuasive, some numerical examples are shown. Numerical results are accord with the theoretical results. This provides a practical approach to construct efficient structure-preserving algorithms multidimensional Maxwell's equations.

A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist several different Hamiltonians introduced for quasi-incompressible elastodynamics based on different multifield variational formulations, there is not much study on the fully incompressible materials in the literature. The adopted mixed formulation can be viewed as a finite-strain generalization of Herrmann variational formulation, and it naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient and scaled mid-point formulas, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. Our analysis and numerical evidence also reveal that the scaled mid-point formula is non-robust numerically. The generalized Taylor–Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano’s formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.

An accurate and practical numerical solver for simulations of shock, vortices and turbulence interaction problems

Numerical simulations of high-speed compressible flows remain challenging in engineering as the appearance of shock waves poses difficulties for high-resolution schemes as well as explicit large eddy simulation methods. Therefore, in this work, we propose a practical numerical solver for simulations of shock-vortex and shock-turbulence problems with the implicit large eddy simulation approach and newly developed low-dissipative schemes. In order to handle the multi-scale raised in the shock-vortex and shock-turbulence problems, it is required that the flow solver can simultaneously solve the large-scale flow structures such as shock waves with numerical oscillation-free, the resolvable flow structures above the grid scale with low-dissipation, and the under-resolve isotropic sub-grid scale with numerical stability. To this end, a low-dissipative, structure-preserving scheme with rigorously adjusted numerical dissipation is employed in the proposed solver. The structure-preserving property of this scheme can ensure that the large-scale structure is solved without numerical oscillations and the low-dissipative property of this scheme can produce high-resolution results for the resolvable flow scales. Moreover, the sub-grid scale is solved and stabilized by the inherent numerical dissipation in the shock-capturing scheme. The proposed numerical solver is then applied to simulate a wide range of shock-vortex and shock-turbulence interaction problems including supersonic planar jets, transonic flows past a deep cavity and impingement of a supersonic jet on a cone mounted on a flat plate. A comparison is also made with the solver using conventional total variation diminishing schemes. The numerical results of the supersonic planar jet have demonstrated that the proposed solver can resolve the small-scale structure such as Kelvin–Helmholtz instability involved in shock and shear layer with high-resolution. Through the results of transonic flow past a deep cavity and comparisons with the experimental data, it is verified that the proposed solver can reproduce the turbulence statistical data. Furthermore, the proposed solver resolves the complex turbulent fluid mechanical phenomena in the impingement of a supersonic jet on a cone, which demonstrates the proposed solver can robustly handle irregular geometry. The proposed solver also enjoys simplicity without using the explicit sub-grid scale model and involving the complexity of very high-order schemes. Thus, this work provides an accurate and practical numerical solver for shock-vortex and shock-turbulence problems in high-speed flows.

High-order, unconditionally maximum-principle preserving finite element method for the Allen–Cahn equation

In this paper, based on the mass-lumping finite element space discretization, we incorporate the integrating factor Runge–Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen–Cahn equation and its conservative forms. The proposed methods are linear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite element method.

Structure-preserving scheme for one dimension and two dimension fractional KGS equations

<abstract><p>In the paper, we study structure-preserving scheme to solve general fractional Klein-Gordon-Schrödinger equations, including one dimension case and two dimension case. First, the high central difference scheme and Crank-Nicolson scheme are used to one dimension fractional Klein-Gordon-Schrödinger equations. We show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate $ O(\tau^2+h^4) $. Moreover, we prove that the resulting scheme can preserve the mass and energy conservation laws. Second, we show Crank-Nicolson scheme for two dimension fractional Klein-Gordon-Schrödinger equations, and the proposed scheme preserves the mass and energy conservation laws in discrete formulations. However, the obtained discrete system is nonlinear system. Then, we show a equivalent form of fractional Klein-Gordon-Schrödinger equations by introducing some new auxiliary variables. The new system is discretized by the high central difference scheme and scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve the energy conservation law. Finally, the numerical experiments including one dimension and two dimension fractional Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.</p></abstract>

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

High-order schemes for the fractional coupled nonlinear Schrödinger equation

<abstract><p>This paper considers the fractional coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the quadratic auxiliary variable method. To this end, we develop the multiple quadratic auxiliary variable approach and then construct a family of structure-preserving schemes with the help of the symplectic Runge-Kutta method for solving the equation. The given schemes have high accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.</p></abstract>