Semi-discrete Formulation Research Articles

A high order numerical method for analysis and simulation of 2D semilinear Sobolev model on polygonal meshes

In this article, we design and analyze a hybrid high-order method for a semilinear Sobolev model on polygonal meshes. The method offers distinct advantages over traditional approaches, demonstrating its capability to achieve higher-order accuracy while reducing the number of unknown coefficients. We derive error estimates for the semi-discrete formulation of the method. Subsequently, these convergence rates are employed in full discretization with the Crank–Nicolson scheme. The method is demonstrated to converge optimally with orders of O(τ2+hk+1) in the energy-type norm and O(τ2+hk+2) in the L2 norm. The reported method is supported by a series of computational tests encompassing linear, semilinear and Allen–Cahn models.

A discontinuous Galerkin finite element method for the Oldroyd model of order one

In this work, we analyze a discontinuous Galerkin finite element method for the equations of motion that arise in the 2D Oldroyd model of order one. We investigate the existence and uniqueness of semidiscrete discontinuous solutions, as well as the consistency of the scheme. We derive new a priori and regularity results for the discrete solution and establish optimal error estimates in ‐norm in time and energy norm in space for the velocity and ‐norm in both time and space for the pressure. Uniform estimates are derived for sufficiently small data. We next apply the backward Euler method to the semidiscrete formulation and establish optimal fully discrete error estimates. At the end, we conduct numerical experiments to support our theoretical results and analyze the findings.

On the complex version of the Cahn–Hilliard–Oono type equation for long interactions phase separation

Abstract This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation.

Finite element analysis of time‐fractional integro‐differential equation of Kirchhoff type for non‐homogeneous materials

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive bound as well as bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of , where is the fractional derivative exponent and and are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.

The virtual element method for the 3D resistive magnetohydrodynamic model

We present a four-field virtual element discretization for the time-dependent resistive magnetohydrodynamics equations in three space dimensions, focusing on the semi-discrete formulation. The proposed method employs general polyhedral meshes and guarantees velocity and magnetic fields that are divergence free up to machine precision. We provide a full convergence analysis under suitable regularity assumptions, which is validated by some numerical tests.

Analysis of algebraic flux correction schemes for semi-discrete advection problems

Based on recent developments regarding the analysis of algebraic flux correction schemes, we consider a locally bound-preserving discretization of the time-dependent advection equation. Specifically, we analyze a monolithic convex limiting scheme based on piecewise (multi-)linear continuous finite elements in the semi-discrete formulation. To stabilize the discretization, we use low order time derivatives in the definition of raw antidiffusive fluxes. Our analytical investigation reveals that their limited counterparts should satisfy a certain compatibility condition. The conducted numerical experiments suggest that this prerequisite is satisfied unless the size of mesh elements is vastly different. We prove global-in-time existence of semi-discrete approximations and derive an a priori error estimate for finite time intervals with a worst-case convergence rate of frac{1}{2} w. r. t. the {textrm{L}}^2 error. This rate is optimal in the setting under consideration because we allow all correction factors of the flux-corrected scheme to become zero. In this case, the algorithm reduces to the bound-preserving discrete upwinding method but the limited counterpart of this scheme converges much faster, in practice. Additional numerical experiments are performed to verify the provable convergence rate for a few variants of the scheme.

Energy-conserving explicit and implicit time integration methods for the multi-dimensional Hermite-DG discretization of the Vlasov-Maxwell equations

We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such invariants. Since explicit RK methods can only ensure preservation of linear invariants but do not provide any control on the system energy, we consider modified explicit RK methods in the family of relaxation Runge-Kutta methods (RRK). These methods can be tuned to preserve the energy invariant at the continuous or semi-discrete level, a distinction that is important when upwind fluxes are used in the discretization of Maxwell's equations since upwind provides a numerical source of energy dissipation that is not present when central fluxes are used. We prove that the proposed methods are able to preserve the energy invariant and to maintain the semi-discrete energy dissipation (if present) according to the discretization of Maxwell's equations. An extensive set of numerical experiments corroborates the theoretical findings. It also suggests that maintaining the semi-discrete energy dissipation when upwind fluxes are used leads to an overall better accuracy of the method relative to using upwind fluxes while forcing exact energy conservation.

Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

This paper formulates the second-order BDF and theta schemes based on an efficient localized meshless technique for computing the approximate solution of the Sobolev model. The model is one of the useful description for modeling the flow of fluids through fractured rocks, movement of moisture in soils. This proposed approach discretizes the nonlinear Sobolev equation in two main steps. In the first step, temporal discretization is accomplished with the help of the second-order BDF and theta schemes, respectively. In the second step, the localized radial basis function partition of unity collocation method (L-RBF-PUCM) is used for spatial discretization, thereby constructing a fully-discrete algorithm. For large linear problems, global techniques have the disadvantage of high computational cost. The proposed method overcomes this shortcoming well and reduces the computational burden by sparsifying the linear system. The convergence analysis and unconditional stability of the semi-discrete formulation are examined by means of the energy method. Numerical results are carried out to verify the validity and accuracy of the L-RBF-PUCM.

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is the reason we call it sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the method’s performance, we solve two common multiscale problems from two models with a long source sequence. The numerical results illustrate that the inversion is accurate and efficient.

Coexistence conditions in generalized discrete-time models of insect population dynamics

Motivated by the annual life histories of insects residing in the temperate-regions of the world, there is a rich tradition of modeling their population dynamics using a discrete-time formalism. Here, we consider an antagonistic interaction between two insect species – a parasitoid wasp that lays an egg within its host insect species. The egg hatches into juvenile parasitoid that develops within the body of the host by using it as resource and this ultimately results in host death. The parasitoid emerges from the host as a free-living insect that goes on to attack other hosts. Such interaction are ubiquitous with over 65,000 known species of parasitoid wasps and such wasps have potential in biologic control of pest insect species. We introduce a general class of discrete-time models for capturing the population dynamics of two competing parasitoid species that attack the same vulnerable stage of the host species. These models are characterized by two density-dependent functions: an escape response defined by the fraction of hosts escaping parasitism; and a competition response defined by the fraction of parasitized hosts that develop into adult parasitoids of either species. Model analysis reveals remarkably simple stability conditions for the coexistence of competing parasitoids. Coexistence occurs, if and only if, the adult host population increases with host reproduction rate, and the log sensitivity of the competition response is less than half. The latter condition implies that any increase in the adult parasitoid population will result in a sufficiently slow increase in the fraction of parasitized hosts that develop into parasitoids of that type. We also consider a semi-discrete formulation of a competing species model that incorporates host density-dependence in the attack rate. This model expands the class of models for which the analytical results apply, and we systematically compare the numerical results of this model to the established coexistence conditions

On Mathematical and Numerical Modelling of Multiphysics Wave Propagation with Polytopal Discontinuous Galerkin Methods: a Review

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of two-dimensional verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods.

Finite element method for nonlocal problems of Kirchhoff-type in domains with moving boundary

This paper is devoted to the analysis of the finite element method for the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equations in a bounded noncylindrical domain with moving boundaries. With the use of the coordinate transformation which fixes the boundaries, the semidiscrete formulation is presented and the convergence and error bounds in the energy norm and for the first order derivative with respect to time in the L2-norm are established. In particular, the error in the energy norm and for the first order derivative with respect to time in the L2-norm is shown to converge with the optimal order O(hr) with respect to the mesh size h and the polynomial degree r≥1. To obtain the fully discrete solution, the generalized-α method is adapted to the semidiscrete formulation. The test problems used are designed to illustrate the behavior of the algorithms. Some numerical tests using Matlab are performed to confirm our theoretical findings.

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

We present an efficient method for the numerical approximation of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method, which not only reduces the number of degrees of freedom compared to the original Virtual Element Method, but also allows for the introduction of an approximation of the nonlinear term that is computable from the degrees of freedom of the discrete solution with a low computational cost, thus significantly improving the efficiency of the method. An error analysis for the semi-discrete formulation is carried out, and an optimal estimate for the error in the L_2-norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with approximations up to order 6.

A hybrid high-order method for Sobolev equation with convection-dominated term

We consider a new hybrid high-order method for the Sobolev equation with convection-dominated term. We adopt the Pk polynomials to approximate the primary unknown on mesh elements and faces where k is a given positive integer. The stability and high accuracy are obtained by using reconstruction to formulate the gradient term and convective term and by adding a suitable stabilization term. We analyze the semi-discrete formulation of the hybrid high-order method and use Crank-Nicolson difference in time to get the corresponding fully-discrete scheme. We prove stability and convergence of the approximate solution in energy norm. The energy errors hold irrespective of the reciprocal of dispersion, diffusion and reaction coefficients. Numerical results underline the efficiency of the proposed methods and our analytical findings.

A reduced unified continuum formulation for vascular fluid–structure interaction

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.

Numerical investigation of abdominal aortic aneurysm hemodynamics using the reduced unified continuum formulation for vascular fluid-structure interaction

We recently demonstrated the reduction of the unified continuum and variational multiscale formulation to a computationally efficient fluid-structure interaction (FSI) formulation via three modeling assumptions pertaining to the vascular wall. Similar to the coupled momentum method introduced by Figueroa et al., the resulting semi-discrete formulation yields a monolithically coupled FSI system posed in an Eulerian frame of reference with only a minor modification of the fluid boundary integral. To achieve uniform second-order temporal accuracy and user-controlled high-frequency algorithmic damping, we adopt the generalized-α method for uniform temporal discretization of the entire coupled system. In conjunction with a fully consistent, segregated predictor multi-corrector algorithm preserving the block structure of the incompressible Navier-Stokes equations in the implicit solver’s associated linear system, a three-level nested block preconditioner is adopted for improved representation of the Schur complement. In this work, we apply our reduced unified continuum formulation to an appropriately prestressed patient-specific abdominal aortic aneurysm and investigate the effects of varying spatial distributions of wall properties on hemodynamic and vascular wall quantities of interest.

A new analysis of Galerkin Legendre spectral methods for coupled hyperbolic/parabolic system arising in unsteady MHD flow of Maxwell fluid and numerical simulation

In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L2 and H1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations.

Domain decomposition and expanded mixed method for parabolic partial differential equations

We developed and analyzed the multiblock mortar expanded mixed method for second order parabolic partial differential equations. This is a domain decomposition method in which the computational domain is expressed as the union of non-overlapping subdomains separated by interfaces. An auxiliary variable is introduced on the interface which represents the pressure and serves as Dirichlet boundary condition for local subdomain problems. The interface variable also plays the part of Lagrange multiplier to enforce flux matching condition on the interfaces. We explored the expanded mixed method to discretize each subdomain. We propose the semi-discrete formulation and address the solvability of the discrete problem. The optimal order convergence is provided for the continuous time case. We also investigate the fully discrete formulation and derived corresponding error estimates. The numerical experiments are conducted to demonstrate the theory developed in the paper.

Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic direction, we also employ a random variable W based on the Q-Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro‐differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.