Conservative Finite Difference Method Research Articles

Soliton solutions of derivative nonlinear Schrödinger equations: Conservative schemes and numerical simulation

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order O(τ2+h2) in the discrete L∞-norm, where h denotes the spatial step size and τ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.

The thermal structure of the Colombian lithosphere: A regional and basin-scale analysis

It is well-established that the thermal state of the lithosphere strongly influences various regional and local geological processes, including crustal deformation, hydrocarbon maturation, hydrogen generation, and geothermal phenomena. Moreover, the thermal structure exhibits high sensitivity to tectonic features, a property of particular significance in Colombia, where three main tectonic plates converge, and lithospheric tearing has been documented. In this contribution, we focus on elucidating the impact of plate architecture on the thermal field in central-eastern Colombia at both shallow and deep levels. To accomplish this, we constructed a series of two-dimensional profiles and derived a numerical solution of the heat equation using the conservative finite difference method. As constraints, we incorporate an inferred distribution of rocks in the deep crust and upper mantle based on global and local lithospheric thickness models. Material parameters for the various rocks, both exposed and inferred, were obtained from the literature. Additionally, we used superficial heat flow estimates and apparent geothermal gradients compiled by the Colombian Geological Survey.Our results suggest a significant influence of the lithospheric Caldas tear on the thermal state of Colombia, with the breaking off occurring in the Nazca plate under the Eastern Cordillera range around 5°N. The modeled asthenospheric heat flow remains approximately 25 mWm−2, except in the northern Eastern Cordillera range, where the background heat flow increases rapidly to 40 mWm−2. Consequently, our model predicts partial melting in the lower crust to the north and a thermally unstable lower crust to the south of the Caldas tear. The material parameters that best fit the surface data suggest the presence of a basement moderately enriched in radioactive elements in the Eastern Llanos basin. After accounting for compaction, we also confirm a strong influence of the tectonic setting on the thermal state of sedimentary basins.

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Hermes-3: Multi-component plasma simulations with BOUT++

A new open source tool for fluid simulation of multi-component plasmas is presented, based on a flexible software design that is applicable to scientific simulations in a wide range of fields. Hermes-3 is built on plasma simulation framework BOUT++, consolidating earlier SD1D and Hermes models into a single code that can be configured at run-time to solve plasma models in 1D, 2D or 3D, either for transport (steady-state) or turbulent (time-evolving) problems, with an arbitrary number of ion and neutral species. We describe the improved numerical algorithms and software design that have been implemented in Hermes-3.To demonstrate the capabilities of this tool, applications relevant to the boundary of tokamak plasmas are presented: 1D simulations of diveror plasmas evolving equations for all charge states of neon and deuterium; 2D transport simulations of tokamak equilibria in single-null X-point geometry with plasma ion and neutral atom species; and simulations of the time-dependent propagation of plasma filaments (blobs).Hermes-3 is publicly available on Github under the GPL-3 open source license. The repository includes documentation and a suite of unit, integrated and convergence tests. Program summaryProgram Title: Hermes-3CPC Library link to program files:https://doi.org/10.17632/rtkvc3r743.1Developer's repository link:https://github.com/bendudson/hermes-3Licensing provisions: GPLv3Programming language: C++14Nature of problem: Simulation of dynamics and steady-state solutions of multiple ion and neutral species in magnetised plasmas using drift-reduced fluid plasma models in 1 to 3 spatial dimensions. Focused on but not restricted to the simulation of turbulence in the boundary of tokamaks.Solution method: Hermes-3 implements a system of flexible software components, that are configured at run-time and coupled by passing a state object between them (the Encapsulate State and Command patterns). Hermes-3 builds on BOUT++ [1], employing the Method of Lines with implicit and explicit time-integration methods in curvilinear coordinates on block-structured meshes, making use of libraries including SUNDIALS, PETSc and Hypre. Hermes-3 implements drift-reduced plasma fluid equations using conservative finite difference methods, and atomic processes that couple plasma and neutral fluids.

Conservative second-order finite difference method for Camassa–Holm equation with periodic boundary condition

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation.

A conservative splitting high-order finite difference method for coupled Gross–Pitaevskii equations in 2D

A conservative splitting high-order finite difference method for coupled Gross–Pitaevskii equations in 2D

Influences of conservative and non-conservative Lorentz forces on energy conservation properties for incompressible magnetohydrodynamic flows

In the analysis of magnetohydrodynamic (MHD) flow, the Lorentz force significantly affects energy properties because the work generated by the Lorentz force changes the kinetic and magnetic energies. Therefore, the Lorentz force and energy conversion should be predicted accurately. Some energy conservation schemes have been proposed and validated. However, the influences of the Lorentz force discretization on conservation and conversion of energy have not yet been clarified. In this study, a conservative finite difference method is constructed for incompressible MHD flows considering the induced magnetic field. We compare the difference in energy conservation properties among three methods of calculating the Lorentz force. The Lorentz forces are calculated in conservative and non-conservative forms, and both compact and wide-range interpolations of magnetic flux density are used to calculate the non-conservative Lorentz force. The compact interpolation method proposed in this study can perform conversions between conservative and non-conservative forms of the Lorentz force even when using the finite difference method. The present numerical method improves the conservation of transport quantity. Five models were analyzed, and the accuracy and convergence of the present numerical method were verified. From the viewpoint of the conservation of the total energy in an ideal inviscid periodic MHD flow, we consider that the calculation using compact interpolation for the Lorentz force is appropriate. This method preserves the total energy even on non-uniform grids. Moreover, the divergence-free condition of the magnetic flux density is discretely satisfied even without the correction of the magnetic flux density. The present numerical method can capture the Hartmann layer in the propagation of an Alfvén wave and accurately predict the tendency of energy attenuation in the analysis of a Taylor decaying vortex under magnetic fields. Analysis of the Orszag–Tang vortex reveals energy dissipation processes and the generation of high current densities. The present numerical method has excellent energy conservation properties and can accurately predict energy conversion. Therefore, this method can contribute to understanding complex unsteady MHD flows.

Convection linearization of energy conservative implicit method for incompressible turbulence simulations

Time accurate implicit methods for unsteady turbulent flows are proposed based on the energy conservative finite difference method of the incompressible Navier-Stokes equations with convection linearization. The standard and Runge-Kutta type high-order temporal linearizations are introduced for the convective term. The time increment independent solution of turbulence is explored for the linearized coupled governing equations. Coupled and uncoupled (splitting) algorithms are then considered to isolate the effect of splitting error. The simulation examples are periodic inviscid, Taylor-Green vortex, and DNS of turbulent plane channel flows. These are test cases for the energy conservation property, the order of accuracy, and the time increment independence of turbulence statistics, respectively. The second order accurate two-stage convection linearized semi-implicit Runge-Kutta (SIRK2L) method works very well and has almost the same conservation property as the fully (spatio-temporally) conservative nonlinear method based on the implicit midpoint (IM) time marching. The second place is the standard third order convection linearization (IM3L) method. The second order convection linearization method (IM2L) is not recommended, although it may be preferred by the point of view of formal accuracy. The effect of splitting error is significant, and the numbers of coupling iteration of 5 and 10 are required for SIRK2L and IM3L methods with the uncoupled (splitting) algorithm, respectively, for the turbulent plane channel flow simulation. Finally, the DNS of forward-facing step flow is carried out as an example of practical problems.

Conservative finite difference methods for the Boussinesq paradigm equation

In this paper, we concentrate on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, by introducing the auxiliary potential function ∂u∂t=Δv and lim|x|→∞v=0, the BPE is reformulated as an equivalent system of coupled equations. Then a class of efficient difference schemes are proposed for solving the resulting system, where one proposed scheme is a two-level nonlinear difference scheme and the other is a three-level linearized difference scheme using the invariant energy quadratization technique. Subsequently, we present the theoretical analysis of the proposed energy conservative finite difference schemes, which involves the discrete energy conservation properties, unique solvability and optimal error estimates. By using the discrete energy method, it is proven that the proposed schemes can achieve the optimal convergence rates of O(Δt2+hx2+hy2) in discrete L2-, H1- and L∞-norms. At last, numerical experiments illustrate the physical behaviors and efficiency of the proposed schemes.

An efficient bound-preserving and energy stable algorithm for compressible gas flow in porous media

Due to the great significance of the natural gas and shale gas, it is becoming increasingly important to simulate compressible gas flow in porous media. The model obeys an energy dissipation law as well as molar density must be positive and bounded in terms of the equation of state. For the purpose of eliminating nonphysical solutions as well as improving the stability in practical simulation, preservation of these properties is essential for a promising numerical method, but it is actually challenging due to the strong nonlinearity and complexity of the model. In this paper, we propose an efficient linearized numerical scheme that inherits the energy dissipation law as well as preserves the boundedness of molar density. Specifically, to treat the logarithmic type Helmholtz free energy density determined by the Peng-Robinson equation of state, we propose a novel adaptive stabilization approach involving the second derivatives of the convex energy terms. At each time step, the stabilization parameter is adaptively updated by a simple and explicit formula to ensure the energy dissipation law. The stabilized and linearized chemical potential allows to formulate the local mass conservation equation as an equivalent convection-diffusion form, and from this, an adaptive time step strategy is proposed to preserve the positivity and boundedness of molar density. The calculation of the time step size is fully explicit and easy to implement. Additionally, the fully discrete scheme is constructed using the conservative cell-centered finite difference method with the upwind strategy, and thus, it enjoys the local mass conservation. Numerical results are also presented to demonstrate the excellent performance of the proposed scheme.

Time second‐order splitting conservative difference scheme for nonlinear fractional Schrödinger equation

This paper concentrates on the error estimation of novel time second‐order splitting conservative finite difference method (FDM) for high‐dimensional nonlinear fractional Schrödinger equation rigorously. The discrete preservation property of our scheme is exhibited. By virtue of the cut‐off technique and discrete energy method, it is shown that our scheme possesses the accuracy of in sense of ‐ norm. Numerical experiments are exhibited to validate the accuracy and conservation property of our scheme.

Unconditional optimal error estimates of conservative methods for Klein–Gordon–Dirac system in two dimensions

In this work, we propose and analyze finite difference methods for solving two-dimensional Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling, it is a great challenge to design and analyze numerical methods for KGD system. To overcome this difficulty, two linearized, decoupled and conservative finite difference methods are presented, which are mass- and energy-conserved. By rigorous error estimates, the conservative methods converge with second-order accuracy in both spatial and temporal discretizations without any requirements on the grid ratios. Several numerical experiments are carried out to illustrate the performance of the proposed numerical methods.

Positive and Conservative Characteristic Block-Centered Finite Difference Methods for Convection Dominated Diffusion Equations

Positive and Conservative Characteristic Block-Centered Finite Difference Methods for Convection Dominated Diffusion Equations

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter 0 < ε ≪ 1 inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at O(ε), while in space, it is at O(1) with the wave speed at O(ε− 1). We adopt one leap-frog, two semi-implicit, and one conservative Crank-Nicolson finite difference methods to numerically discretize the Dirac equation in one dimension and establish rigorously the error estimates which depend explicitly on the time step τ, mesh size h, and the small parameter ε. The error bounds indicate that, to obtain the “correct” numerical solution in the massless and nonrelativistic regime, i.e., 0 < ε ≪ 1, all these finite difference methods share the same ε-scalability as time step τ = O(ε3/2) and mesh size h = O(ε1/2). A large number of numerical results are reported to verify the error estimates.

Structure preserving — Field directional splitting difference methods for nonlinear Schrödinger systems

A computational framework of high order conservative finite difference methods to approximate the solution of a general system of N coupled nonlinear Schrödinger equations (N-CNLS) is proposed. Exact conservation of the discrete analogues of the mass and the system’s Hamiltonian is achieved by decomposing the original system into a sequence of smaller nonlinear problems, associated to each component of the complex field, and a modified Crank–Nicolson time marching scheme appropriately designed for systems. For a particular model problem, we formally prove that a method, based on the standard second order difference formula, converges with order τ+h2; and, using the theory of composition method, schemes of order τ2+h2 and τ4+h2 are derived. The methodology can be easily extended to other high order finite difference formulas and composition methods. Conservation and accuracy are numerically validated.

Mass conservative characteristic finite difference method for convection–diffusion equations

In the paper, the space second-order conservative characteristic finite difference method for solving convection–diffusion equations is developed and analysed. Applying the directional derivative and mass correction technique, the convection–diffusion equations are changed into the parabolic equations, where the convection term and unsteady term are considered as one term. The splitting implicit solution-flux coupled scheme computes the solutions and fluxes on the staggered meshes. The mass conservation, stability and convergence of our proposed characteristic difference method are analysed theoretically. Numerical experiments are presented to illustrate mass conservation and convergence.

A sharp-interface immersed boundary method for moving objects in compressible viscous flows

Sharp-interface Immersed Boundary Methods for moving solids in compressible viscous flows often exhibit spurious noise propagating from the moving boundary. In compressible flows, filtering or upwinding discretization techniques are often used to overcome the problem. In the present work, we show that using a conservative energy-preserving finite difference method for convection in combination with a Ghost-Point-Forcing-Method (GPFM) is able to keep controlled the pressure-velocity spurious noise. In order to deal with compressible flow in a wide range of Mach numbers the shock-dynamics appearing in high-speed flows conditions has been addressed hybridising the latter scheme with a fifth-order weighted-essentially-non-oscillatory (WENO) scheme. The latter, in the path of keeping the numerical dissipation minimal, has been confined around the shock locations using a proper detector. In this respect, the method appears a suitable alternative for direct and large-eddy simulation of moving objects in compressible flows. The entire methodology was found to be robust ranging from weakly compressible to highly supersonic flows including shocks. In order to prove the cleanliness and the robustness of the method, several well-documented benchmarks and test cases, in a wide range of both Reynolds and Mach numbers, are presented.

Regularized numerical methods for the logarithmic Schrödinger equation

We present and analyze two numerical methods for the logarithmic Schr{\"o}dinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schr{\"o}dinger equation (RLogSE) with a small regularized parameter 0 < $\epsilon$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence rate O($\epsilon$). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O($\tau$ 1/2 ln($\epsilon$ --1)) with $\tau$ > 0 the time step, which implies an error bound at O($\epsilon$ + $\tau$ 1/2 ln($\epsilon$ --1)) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.

Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer framework

In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.