Convergence and structure theorems for order-preserving dynamical systems with mass conservation

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

$i=1,$ $\cdots,$ $d$ , under the assumption that $f_{i}$ satisfies the Caratheodory condition, where $x(t)$ stands for $(x_{1}(t), \cdots, x_{d}(t))$ and prime denotes differentiation with respect to $t$ . We assume the existence of a finite number $\alpha$ such that for each $j=2,$ $\cdots,$ $m,$ $\alpha\leqq g_{j}(t, x)\leqq t$ , whenever $g_{j}(t, x)$ is defined; the delays $t-g_{j}(t, x)$ may be unbounded. This type of system arises in studying a two-body problem of classical electrodynamics $[6, 7]$ . Driver [4] developed the basic theory (existence, uniqueness and dependence of solutions, etc.) for the initial value problem for delay differential equations (E) with continuous $f_{i}[5]$ . Since then the theory of delay differential equations (E) has been studied by many authors. Among them Bullock [1] showed the existence theorem and uniqueness theorem for delay differential equations (E’) of Caratheodory type. On the other hand, Strauss and Yorke [13] constructed a fundamental theory for ordinary differential equations by using the convergence theorem which is a generalization of Kamke’s theorem (see [8], Theorem 3.2). Their method proves to be very important in studying the fundamental theory of functional (or delay) differential equations. Costello [3] extended their results to functional differential equations of Caratheodory type with finite delay,

A variant of the flux-form semi-Lagrangian (FFSL) advection scheme is presented suitable for use on arbitrary unstructured meshes. The time-step used with the scheme is not constrained by the Courant number, resulting in model run-times that may be many times faster than Courant-constrained schemes. The scheme also has the property of mass conservation. This makes the scheme a potential candidate for offline simulation of tracers in unstructured sediment transport, biogeochemical or atmospheric chemistry models. We describe the numerics of the scheme and apply it to a number of idealised and real applications to demonstrate its utility. In these test cases we compare results from the offline FFSL scheme with those generated by the original hydrodynamic model which uses high order conservative tracer transport. We assess the FFSL scheme for accuracy, conservation and computational efficiency. Additionally, traditional semi-Lagrangian advection schemes are included in assessments for comparison. Results indicate that the FFSL scheme produces closer agreement with tracer distributions generated by the hydrodynamic model than the semi-Lagrange scheme, and mass conservation was vastly superior. The FFSL scheme was also an order of magnitude faster than the hydrodynamic model in certain cases. The FFSL scheme presented here therefore offers a viable mass-conserving, computationally-efficient option for use in offline transport models on arbitrary unstructured meshes.

A comparative study of local and nonlocal Allen-Cahn equations with mass conservation

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

On mass and momentum conservation in the variable-parameter Muskingum method

It is well known that the Cahn-Hilliard equation satisfies the energy dissipation law and the mass conservation property. Recently, the radial basis function-finite difference (RBF-FD) approach and its numerous variants have garnered significant attention for the numerical solution of surface-related problems, owing to their intrinsic advantage in handling complex geometries. However, existing RBF-FD schemes generally fail to preserve mass conservation when solving the Cahn-Hilliard equation on smooth closed surfaces. In this paper, based on an L2 projection method, two numerically efficient RBF-FD schemes are proposed to achieve mass-conservative numerical solutions, which are demonstrated to preserve the mass conservation law under relatively mild time-step constraints. Spatial discretization is performed using the RBF-FD method, while based on the convex splitting method and a linear stabilization technique, the first-order backward Euler formula (BDF1) and the second-order Crank-Nicolson (CN) scheme are employed for temporal integration. Extensive numerical experiments not only validate the performance of the proposed numerical schemes but also demonstrate their ability to utilize mild time steps for long-term phase-separation simulations.

Abstract. The semi-Lagrangian absolute vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at the Hydrometeorological Centre of Russia. The distinct feature of the SL-AV dynamical core is the semi-implicit, semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. A semi-implicit, semi-Lagrangian approach allows for long time steps but violates the global and local mass conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no a posteriori mass-fixing algorithm is applied. However, the global mass-fixing algorithms degrade the local mass conservation. The new inherently mass-conservative version of the SL-AV model dynamical core presented here ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume, semi-Lagrangian discretization for a continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). Numerical experiments show that the new version of the SL-AV dynamical core presented combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain-induced Rossby-wave test and baroclinic instability test for the mass-conservative dynamical core are found to be in agreement with the results available in the literature.

Finite element approaches generally do not guarantee exact satisfaction of conservation laws especially when Dirichlet-type boundary conditions are imposed. This article discusses improvement of the global mass conservation property of quasi-bubble finite element solutions for the shallow water equations, focusing on implementations of the surface-elevation boundary conditions. We propose two alternative implementations, which are shown by numerical verification to be effective in improving the smoothness of solutions near the boundary and in reducing the mass conservation error. The improvement of the mass conservation property contributes to augmenting the reliability and robustness of long-term time integrations. Copyright © 2006 John Wiley & Sons, Ltd.

We propose in this paper a time second order mass conservative algorithm for solving advection–diffusion equations. A conservative interpolation and a continuous discrete flux are coupled to the characteristic finite difference method, which enables using large time step size in computation. The advection–diffusion equations are first transformed to the characteristic form, for which the integration over the irregular tracking cells at previous time level is proposed to be computed using conservative interpolation. In order to get second order in time solution, we treat the diffusion terms by taking the average along the characteristics and use high order accurate discrete flux that are continuous at tracking cell boundaries to obtain mass conservative solution. We demonstrate the second order temporal and spatial accuracy, as well as mass conservation property by comparing results with exact solutions. Comparisons with standard characteristic finite difference methods show the excellent performance of our method that it can get much more stable and accurate solutions and avoid non-physical numerical oscillation.

On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method

This paper studied the mass conservation and constancy-preserving conditions of the advection scheme in the Hydrological Simulation Program-Fortran (HSPF) model. Equations were derived to examine the conditions for achieving both mass conservation and constancy-preserving properties. Case studies of simulating conservative materials, water temperature, and dissolved oxygen were conducted to show the impacts of the advection scheme on model applications. The mass conservative condition of HSPF can only be achieved weakly even though the advection equation is written in a mass conserved format for the current advection scheme. The constancy-preserving condition is strongly violated due to the inconsistency with continuity. The advection scheme was modified by the writer to use one single weighting factor for solving both water routing and material advection. Unrealistic results are removed and mass conservation and constancy-preserving conditions are achieved with the modified advection scheme.

Mass conservative, positive definite integrator for atmospheric chemical dynamics

In this paper, we develop Galerkin approximations for determining the stability of delay differential equations (DDEs) with time periodic coefficients and time periodic delays. Using a transformation, we convert the DDE into a partial differential equation (PDE) along with a boundary condition (BC). The PDE and BC we obtain have time periodic coefficients. The PDE is discretized into a system of ordinary differential equations (ODEs) using the Galerkin method with Legendre polynomials as the basis functions. The BC is imposed using the tau method. The resulting ODEs are time periodic in nature; thus, we resort to Floquet theory to determine the stability of the ODEs. We show through several numerical examples that the stability charts obtained from the Galerkin method agree closely with those obtained from direct numerical simulations.

Aim of the paper is to obtain numerical solution of some nonlinear delay differential equations (NDDEs) using Iterative scheme. Banach contraction method (BCM) is investigated on some nonlinear delay differential equations to find their solutions. Numerical results and convergence theorem are presented, and error analysis is discussed for some delay differential equations to show that proposed method is suitable for solving NDDEs. The BCM reduces complex calculations, avoids discretization, linearization, perturbation and save calculation time.