Semi-implicit Time Scheme Research Articles

Existence and uniqueness for the evolutionary impulse control problem using an asynchronous algorithms

Impulse control problems pose significant challenges in decision making under uncertainty, particularly in scenarios where actions must be taken at irregular intervals or in response to abrupt changes in the environment. Ensuring both the existence and uniqueness of optimal control policies in such settings is paramount for practical implementation and theoretical understanding. In this study, we investigate the existence and uniqueness of solutions for impulse control problems utilizing asynchronous algorithms. This study presents a novel demonstration concerning the uniqueness and existence of Partially Quasi-Variational Inequalities (PQVIs). Our approach unfolds through a four-step methodology, leveraging distinctive features of a discrete iterative technique. This methodology integrates semi-implicit techniques with respect to the variable t and employs spatial approximation via finite element methods. We establish a pivotal connection between a fixed-point mapping and the discrete EQVIs system, utilizing it to formulate a discrete algorithm featuring a semi-implicit time scheme. Furthermore, we introduce a monotone iterative scheme inspired by Bensoussan’s algorithm and rigorously confirm several of its properties through mathematical proofs. This work contributes to advancing the understanding and application of iterative techniques in solving PQVIs, offering insights into their uniqueness and existence.

New error analysis and recovery technique of a class of fully discrete finite element methods for the dynamical inductionless MHD equations

In this paper, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic (MHD) equations. The methods use the semi-implicit backward Euler scheme in time and use the standard inf–sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L∞(0,T;L2(Ω))-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor–Hood element supplies the optimal accuracy for the velocity in L2(0,T;H1(Ω))-norm and the pressure in L1(0,T;L2(Ω))-norm. Moreover, we propose a simple recovery technique to obtain a new numerical current density of first-order higher accuracy in the spatial direction by re-solving a mixed Poisson equation. Numerical experiments are performed to verify the theoretical analysis.

Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Abstract A set of control parameters is introduced in the fully elastic nonhydrostatic Euler equations formulated in the mass-based vertical coordinate of Laprise. Contrary to the classical approach, the hydrostatic limit is represented by a subspace of control parameters, instead of a single point. By finding a suitable path from the fully compressible equations to the hydrostatic subspace, we are able to construct a blended system with acoustic modes slowed down and gravity modes nearly unaffected. Numerical stability of the discretized system is thus improved, and the solution remains essentially the fully compressible one. Alternatively, control parameters can be used to redefine the linear model of the constant coefficients semi-implicit time scheme, increasing the numerical stability of the fully compressible system. With a careful choice of the control parameters in both, the linear model used in the semi-implicit temporal scheme, and in the full model, the blended system does not deteriorate the compressible solution while its semi-implicit temporal discretization is more stable. We illustrate the potential of the method in several simple examples and in real case studies using the ALADIN system.

Data accompanying the paper "Stability properties of the constant coefficients semi-implicit time schemes solving non-filtering approximation of the fully compressible equations"

Data accompanying the paper "Stability properties of the constant coefficients semi-implicit time schemes solving non-filtering approximation of the fully compressible equations"

Data accompanying the paper "Stability properties of the constant coefficients semi-implicit time schemes solving non-filtering approximation of the fully compressible equations"

Data accompanying the paper "Stability properties of the constant coefficients semi-implicit time schemes solving non-filtering approximation of the fully compressible equations"

Comparison of Two Aspects of a PDE Model for Biological Network Formation

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor C and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector m∈R2 and four linear systems in the case of the symmetric conductivity tensor C∈R2×2 at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems.

Magnetized fluid electron model within a two-dimensional hybrid simulation code for electrodeless plasma thrusters

An axisymmetric fluid model for weakly-collisional, magnetized electrons is introduced and coupled to a particle-in-cell model for heavy species to simulate electrodeless plasma thrusters. The numerical treatment of the model is based on a semi-implicit time scheme, and specific algorithms for solving on a magnetic field aligned mesh. Simulation results of the plasma transport are obtained for a virtual electrodeless thruster. The particle and energy fluxes of electrons are discussed. A first phenomenological model is included for the anomalous cross-field electron transport, and a second one for the anomalous parallel-field electron cooling in the plume. The balances of the plasma properties reveal that wall losses are the crucial reason for the poor thrust efficiency of these thrusters. The magnetic thrust inside the source could be negative and largely depending on the location of the magnetic throat, which is found uncoupled from the location of the plasma beam sonic surface. Furthermore, a sensitivity analysis of the results against the simulated plume extension shows that finite plumes imply an incomplete electron expansion, which leads to underestimating the performances.

A nonlinear model for stage-structured population dynamics with nonlocal density-dependent regulation: An application to the fall armyworm moth

The assessment and the management of the risks linked to insect pests can be supported by the use of physiologically-based demographic models. These models are useful in population ecology to simulate the dynamics of stage-structured populations, by means of functions (e.g., development, mortality and fecundity rate functions) realistically representing the nonlinear individuals physiological responses to environmental forcing variables. Since density-dependent responses are important regulating factors in population dynamics, we propose a nonlinear physiologically-based Kolmogorov model describing the dynamics of a stage-structured population in which a time-dependent mortality rate is coupled with a nonlocal density-dependent term. We prove existence and uniqueness of the solution for this resulting highly nonlinear partial differential equation. Then, the equation is discretized by finite volumes in space and semi-implicit backward Euler scheme in time. The model is applied for simulating the population dynamics of the fall armyworm moth (Spodoptera frugiperda), a highly invasive pest threatening agriculture worldwide.

A divergence-free generalized moving least squares approximation with its application

An approximation based on moving least squares for vector-valued functions satisfying the divergence-free property was employed in [67] by Trask, Maxey, and Hu. They changed the traditional polynomial bases in moving least squares approximation to the vector-valued polynomial basis functions satisfying the divergence-free property for constructing a new vector-valued approximation. In this paper, we adopt [67], but we consider another approach, i.e., a direct method based on the generalized moving least squares (GMLS) approximation for vector-valued functions, which satisfy the divergence-free property. This GMLS approximation produces diffuse or uncertain derivatives for each component of the proposed vector-valued approximation satisfying the divergence-free property. Using the new approach, the linear functionals such as the derivatives act only on each row of the vector-valued polynomial basis functions, which reduces the computational cost comparing with the divergence-free moving least squares approximation. The pointwise error estimates of the presented approximation are obtained for the bounded sub-domains in Rd(d≥2) with an interior cone condition. Some numerical results are provided to confirm the obtained theoretical results. As an application, the Cahn-Hilliard-Hele-Shaw (CHHS) equation is solved numerically via a stabilized semi-implicit scheme in time that holds the mass conservative and energy dissipative properties and also the GMLS approximation in space. Besides, the developed vector-valued approximation has been used to compute the Leray-Helmholtz projection of the advective velocity.

The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations

PurposeThis study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.Design/methodology/approachFirst, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.FindingsThis study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.Originality/valueThis study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

Simulations of Dynamical Gas–Dust Circumstellar Disks: Going Beyond the Epstein Regime

In circumstellar disks, the size of dust particles varies from submicron to several centimeters, while planetesimals have sizes of hundreds of kilometers. Therefore, various regimes for the aerodynamic drag between solid bodies and gas can be realized in these disks, depending on the grain sizes and velocities: Epstein, Stokes, and Newton, as well as transitional regimes between them. For small bodies moving in the Epstein regime, the time required to establish the constant relative velocity between the gas and bodies can be much less than the dynamical time scale for the problem - the time for the rotation of the disk about the central body. In addition, the dust may be concentrated in individual regions of the disk, making it necessary to take into account the transfer of momentum between the dust and gas. It is shown that, for a system of equations for gas and monodisperse dust, a semi-implicit first-order approximation scheme in time in which the interphase interaction is calculated implicitly, while other forces, such as the pressure gradient and gravity are calculated explicitly, is suitable for stiff problems with intense interphase interactions and for computations of the drag in non-linear regimes. The piece-wise drag coefficient widely used in astrophysical simulations has a discontinuity at some values of the Mach and Knudsen numbers that are realized in a circumstellar disk. A continuous drag coefficient is presented, which corresponds to experimental dependences obtained for various drag regimes.

A Large Time-Stepping Mixed Finite Method of the Modified Cahn–Hilliard Equation

The main purpose of this paper is to solve the modified Cahn–Hilliard equation via a large time-stepping mixed finite-element method to ease the time-stepping limit caused by small parameters and nonlinear terms. The modified Cahn–Hilliard equation is discretized by mixed finite-element method in space and first-order semi-implicit scheme in time. The energy stability and error analysis of the fully discrete semi-implicit scheme are proved. Finally, a series of numerical experiments are presented to verify the conclusion of theoretical part.

Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation

The main objective of this paper is to study the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. We prove the global attractor and stationary statistical properties of the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation converge to those of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation as the time step goes to zero.

Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation

In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.

Numerical investigation and correlations for heat diffusion through planar ablative thermal protection systems

In the present study, heat diffusion through a planar ablative Thermal Protection System (TPS) is numerically investigated by modeling the problem as one-dimensional transient heat conduction equation in Cartesian coordinates subject to the adiabatic back wall and aerodynamic heating on the other surface. The surface exposed to aerodynamic heating undergoes sensible heating until the surface temperature reaches an ablative temperature of the material. Further exposure of the material to heat flux results in material getting ablated. Ablation is modeled as Stefan-type wherein layers of material are immediately removed upon melt after reaching ablative temperature. Boundary immobilization method is used to fix the moving boundary and the governing equations are solved using finite difference scheme in space and Crank-Nicolson semi-implicit scheme in time, after expressing them in non-dimensional form. A FORTRAN code is developed to solve the set of equations using Tri-Diagonal Matrix Algorithm (TDMA). Parametric studies are conducted and new correlations are developed for predicting the amount of material ablated as a function of non-dimensional heat flux, Stefan number and non-dimensional time. Correlations are also developed to predict the non-dimensional time when back-wall that protects the vehicle interiors from extreme heat flux environment attains non-dimensional temperature 0.1. Results show that the developed correlations predict the parameter very well and errors are within acceptable limits.

$$L^{\infty }$$ L ∞ -error estimate of a parabolic quasi-variational inequalities systems related to management of energy production problems via the subsolution concept

The main propose of this paper is to provide an error estimate on uniform norm of the parabolic quasi-variational inequalities system related to management of energy production problems, using semi-implicit time scheme with Galerkin spatial methods. Moreover, a new proof of the existence and uniqueness of the solution are given by the introduction of a constructive presented algorithm. Furthermore, an optimally \(L^{\infty }\)-asymptotic behavior in maximum norm is given. The approach is based on the subsolution concept and discrete regularity.

On finite element approximation of system of parabolic quasi-variational inequalities related to stochastic control problems

In this paper, an optimal error estimate for system of parabolic quasi-variational inequalities related to stochastic control problems is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L∞-asymptotic behavior in maximum norm is proved using the semi-implicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolution and discrete regularity.

An Optimal L∞–error Estimate for an Approximation of a Parabolic Variational Inequality

ABSTRACTIn this article, an optimal error estimate for parabolic variational inequalities is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L∞-asymptotic behavior in uniform norm is proved using the semi-implicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolutions.

Computational gas dynamics in a wide range of Mach number on heterogeneous cluster architecture

The specific features of numerical method for gas dynamics problems are considered in this paper. The viscous flows are considered in a wide range of Mach number. The numerical method is based on conservation laws applied on unstructured heterogeneous mesh with further integration on complex cluster architectures. Application of unstructured mesh allows one to perform calculation of geometrically complex 3D objects with strict conservation properties. The developed numerical method includes multiple flux solvers (Godunov's, HLLC, Roe-Fix, AUSM, etc), first and second order spatial approximation, and explicit, semi-implicit or implicit first order time schemes. The method utilizes heterogeneous GPU HPC clusters. A wide range of problems is used for verification of the developed computational complex. All results on these problems had a good agreement with reference data.

Parallel multilevel solvers for the cardiac electro-mechanical coupling

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.