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In this paper, we develop a fast linearized virtual element method (VEM) for the approximation of the nonlinear time-fractional diffusion equations on polygonal meshes. The L1-scheme with graded meshes is used to deal with the non-smooth system, the Newton linearized method is adopted to handle the nonlinear term and VEM is employed to discrete the spatial variable. Then the error splitting approach is used to prove the unconditional optimal error estimate of the fully discrete linearized L1-VEM scheme. In order to reduce the storage and computational cost caused by the nonlocality of the Caputo fractional operator, a fast memory-saving L1-VEM is developed. It is proved that the difference between the solution of the L1-VEM and the fast L1-VEM can be made arbitrarily small and is independently of the sizes of the time and/or space grids. Finally, numerical results are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number N of samples used to discretized the probability space. We show that this reduced system can be solved with optimal O(N) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L 1 -norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest. [ABSTRACT FROM AUTHOR]

A novel framework is proposed in this research based on multilevel method to solve the optimal control problem. In recent dacades, the mathematical theory of optimal control has rapidly developed into an important and separate field of applied mathematics. The solution of nonlinear partial differential equations is considerably difficult, and the theory of their optimal control is still an open field in many respects. These optimization problems have found diverse applications in various sciences including electrical engineering, mechanical engineering, and aerospace. Current methods for solving this class of optimal control problems usually fall into two classes: discrete-then-optimization or optimization-then-discrete approaches. The proposed approach, however, does not require discretization as it involves rewriting the optimal control problem as a multi-objective optimization problem followed by its solution with a feedforward single-layer artificial neural network based on learning through by the multi-level Levenberg–Marquardt method. Moreover, the convergence of the approach was discussed and some numerical results are presented. [ABSTRACT FROM AUTHOR]

We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov–Poincaré operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements. [ABSTRACT FROM AUTHOR]

This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows. [ABSTRACT FROM AUTHOR]

In this paper, we describe an upgrade of the Alya code with up-to-date parallel linear solvers capable of achieving reliability, efficiency and scalability in the computation of the pressure field at each time step of the numerical procedure for solving a Large Eddy Simulation formulation of the incompressible Navier–Stokes equations. We developed a software module in the Alya's kernel to interface the libraries included in the current version of PSCToolkit, a framework for the iterative solution of sparse linear systems, on parallel distributed-memory computers, by Krylov methods coupled to Algebraic MultiGrid preconditioners. The Toolkit has undergone various extensions within the EoCoE-II project with the primary goal of facing the exascale challenge. Results on a realistic benchmark for airflow simulations in wind farm applications show that the PSCToolkit solvers significantly outperform the original versions of the Conjugate Gradient method available in the Alya's kernel in terms of scalability and parallel efficiency and represent a very promising software layer to move the Alya code toward exascale. [ABSTRACT FROM AUTHOR]

This research article introduces a novel approach based on the fuzzy Adomian decomposition method (FADM) to solve specific time fuzzy fractional partial differential equations with initial and boundary conditions (IBCs). The proposed approach addresses the challenge of incorporating both initial and boundary conditions into the FADM framework by employing a modified approach. This approach iteratively generates a new initial solution using the decomposition method. The method presented here offers a significant contribution to solving fuzzy fractional partial differential equations (FFPDEs) with fuzzy IBCs, a topic that has received limited attention in the literature. Furthermore, it satisfies a high convergence rate with minimal computational complexity, establishing a novel aspect of this research. By providing a series solution with a small number of recursive formulas, this method enhances accuracy and emerges as a preferred choice for tackling FFPDEs with mixed initial and boundary conditions. The effectiveness of the proposed technique is further supported by the inclusion of several illustrative examples. [ABSTRACT FROM AUTHOR]

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomains is accomplished through the utilization of piecewise-linear interpolation. It is shown that the algorithm provides uniformly convergent numerical approximations to the solution. Our analysis utilizes some novel auxiliary problems, barrier functions, and a new maximum principle result. More importantly, we showed that only one iteration is needed for small values of the perturbation parameter. Some numerical results supporting the theory and demonstrating the effectiveness of the algorithm are presented. [ABSTRACT FROM AUTHOR]

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting. [ABSTRACT FROM AUTHOR]

We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom. [ABSTRACT FROM AUTHOR]

We propose a simple numerical procedure to approach the symbol of a self-adjoint linear operator A by using trace estimates of a corresponding discretization matrix A, with numerical data. The Symbol Approximation Method (SAM) is based on an adaptation of the matrix trace estimator to successive distinct numerical spectral bands in order to build a piece-wise constant function as an approximation of the symbol σ of A . The decomposition of the spectral interval into band of frequencies is proposed with several approaches, from the formal spectral to the multi-grid one. We apply the new method to different operators when discretized in finite differences or in finite elements. The SAM is also proposed as a tool for the modeling of waves equations, and is presented a means to capture an additional linear damping term (hidden operator) in hydrodynamics models such as Korteweig–de Vries or Benjamin Ono equations. [ABSTRACT FROM AUTHOR]

The phase-field surfactant model is popular to study the dynamics of surfactant-laden phase separation in a binary mixture. In this work, we numerically investigate the H - 1 -gradient flow based phase-field surfactant mathematical model using an energy dissipation-preserving numerical method. The proposed method adapts a Lagrange multiplier method. The present method not only preserves the unconditional stability, but also satisfies the original energy dissipation law, which is different from the modified energy dissipation laws estimated by the scalar auxiliary variable and invariant energy quadratization methods. An effective scheme is introduced to solve the weakly coupled discrete equations. In one time cycle, we only need to calculate four linear, fully decoupled discrete equations with constant coefficients and compute two nonlinear algebraic equations using Newton's iteration. The computational experiments indicate that the proposed method is accurate and satisfies the original energy stability. Moreover, the long-time behaviors of surfactant-laden phase separation can also be well simulated. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of great importance to develop a robust numerical method for investigating the solution behaviour of the CH equation. We present the formulation of the Neumann–Neumann method applied to the CH equation and study its convergence behaviour in one and two spatial dimension for two subdomains and also extend the method for logarithmic nonlinearity. We also present the nonlinear NN method for the CH equation. We illustrate the theoretical results by providing numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose and present a non-overlapping substructuring-type iterative algorithm for the Cahn–Hilliard (CH) equation, which is a prototype for phase-field models. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of CH equation has. Here we present a formulation for the linear and non-linear Dirichlet–Neumann (DN) methods applied to the CH equation and study the convergence behaviour in one and two spatial dimensions in multiple subdomains. We show numerical experiments to illustrate our theoretical findings and effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal L 2 error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This work provides an inexact primal-dual algorithm for a large class of optimal transport problems. It is based on the implicit Euler discretization of a proper dynamical system for linearly constrained convex optimization problems, and by using the tool of Lyapunov function, the global (super-)linear convergence rate is established for the objective residual and feasibility violation. The presented method contains an inner problem that possesses a strong semismoothness property, which motivates the use of the semismooth Newton iteration. In addition, by exploring the hidden structure of the problem itself, the linear equation arising from the Newton step is transferred equivalently into a graph Laplacian system, for which a robust algebraic multigrid method is proposed and analyzed via the famous Xu–Zikatanov identity. Finally, numerical tests are provided to validate the efficiency of our method. [ABSTRACT FROM AUTHOR]

Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented. [ABSTRACT FROM AUTHOR]

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of compressible fluid flow in a fractured porous medium. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We aim to develop fast-convergent and accurate global-in-time domain decomposition (DD) methods for such a reduced fracture model, in which smaller time step sizes in the fracture can be coupled with larger time step sizes in the subdomains. Using the pressure continuity equation and the tangential PDEs in the fracture-interface as transmission conditions, three different DD formulations are derived; each method leads to a space-time interface problem which is solved iteratively and globally in time. Efficient preconditioners are designed to accelerate the convergence of the iterative methods while preserving the accuracy in time with nonconforming grids. Numerical results for two-dimensional problems with non-immersed and partially immersed fractures are presented to show the improved performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping. [ABSTRACT FROM AUTHOR]

The Murnaghan model of the doubly dispersive equation, which is well-known in the field of materials research, is taken into consideration in this work. This equation is resolved using three different mathematical methods. The analytical method can produce traveling wave solutions by utilizing the wave transform. While bright-dark soliton and bright optical soliton solutions are produced by using auxiliary equation method, hyperbolic type traveling wave solutions are produced with the 1 / G ′ -expansion method. The solutions produced by both analytical methods are different from the literature. A new discussion area is created using traveling wave solutions to this problem, which also has a Conformable derivative operator. It is a cognitive fact that the solutions of partial differential equations shed light on the physical phenomenon. In light of this scientific fact, it comprises scientific debates that take into account the material's density difference, Poisson's ratio, material-specific wave velocity, and laboratory predefined values. These discussions are supported by the solutions obtained by numerical technique. In addition, the accuracy of the results obtained with the numerical technique is analyzed with the L 2 and L ∞ norm errors and the datas have been compared with the table. After discussing the advantages and disadvantages of both mathematical techniques, scientific interpretations that will shed light on physical and chemical phenomena, the effects of parameters on the solution function through traveling wave solutions are discussed and supported by graphics. [ABSTRACT FROM AUTHOR]

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen–Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn–Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an O (t + h 2) accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies. [ABSTRACT FROM AUTHOR]

In this article we propose a scalable shape optimization algorithm which is tailored for large scale problems and geometries represented by hierarchically refined meshes. Weak scalability and grid independent convergence is achieved via a combination of multigrid schemes for the simulation of the PDEs and quasi Newton methods on the optimization side. For this purpose a self-adapting, nonlinear extension operator is proposed within the framework of the method of mappings. This operator is demonstrated to identify critical regions in the reference configuration where geometric singularities have to arise or vanish. Thereby the set of admissible transformations is adapted to the underlying shape optimization situation. The performance of the proposed method is demonstrated for the example of drag minimization of an obstacle within a stationary, incompressible Navier–Stokes flow. [ABSTRACT FROM AUTHOR]

This work proposes a multigrid Waveform Relaxation Method (WRMG) that uses the finite difference method for the discretization to solve Pennes' bioheat equation. There is no evidence in the literature of using the WRMG to solve this equation. The proposed algorithm is based on the red-black Gauss-Seidel smoother in space and the line Gauss-Seidel smoother in time. Verification of code is presented. The performance analysis of the multigrid method confirms the convergence and efficiency of the algorithm. The method, which favors parallel architecture, is tested in two-dimensional numerical experiments on an academic problem and for the thermal analysis of human skin. [ABSTRACT FROM AUTHOR]

We construct and analyze a domain decomposition method to solve a class of singularly perturbed parabolic problems of reaction-diffusion type having Robin boundary conditions. The method considers three subdomains, of which two are finely meshed, and the other is coarsely meshed. The partial differential equation associated with the problem is discretized using the finite difference scheme on each subdomain, while the Robin boundary conditions associated with the problem are approximated using a special finite difference scheme to maintain the accuracy. Then, an iterative algorithm is introduced, where the transmission of information to the neighbours is done using a piecewise linear interpolation. It is proved that the resulting numerical approximations are parameter-uniform and, more interestingly, that the convergence of the iterates is optimal for small values of the perturbation parameters. The numerical results support the theoretical results about convergence. [ABSTRACT FROM AUTHOR]

To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. We also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.

We introduce a domain decomposition method of discrete Schwarz waveform relaxation (DSWR) type for a singularly perturbed nonlinear parabolic problem. The method utilizes Shishkin transition parameter for a space–time decomposition of the computational domain. In each subdomain, the problem is discretized using the central differencing and backward difference schemes on a uniform mesh in space and time directions, respectively. Further, the exchange of information between the subdomains is done through the Dirichlet data that leads to optimal convergence. We analyse the convergence of the developed method and show that the method converges very fast for small perturbation parameter and provides uniformly convergent approximations to the solution of the nonlinear problem. Finally, with some numerical experiments, we illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations in space and time with full upwind, and we show primal and dual consistency. Stability and convergence estimates are provided with respect to a mesh-dependent DG norm which includes the L 2 norm at final time. Numerical experiments confirm that the a priori results are of optimal order also for solutions with low regularity, and we show that the error in the DG norm can be closely approximated with a residual-type error indicator. [ABSTRACT FROM AUTHOR]

The Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is the sum of operators A 1 , A 2 , ... , A m . Each summand is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval the classical difference problems are solved in parallel (independently from each other) with the operators A 1 , A 2 , ... , A m . The weighted average of the obtained solutions is announced as an approximate solution at the right end of the local interval. The convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence. [ABSTRACT FROM AUTHOR]

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations' residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model. [ABSTRACT FROM AUTHOR]

The accuracy of information transmission while solving domain decomposed problems is crucial to the smooth transition of a solution around the interface/overlapping region. This paper describes a systematical study on an accuracy-enhancing interface treatment algorithm based on the back and forth error compensation and correction method (BFECC). By repetitively employing a low order interpolation technique (usually local 2nd order) 3 times, this algorithm achieves local 3rd order accuracy. Analytical derivation for any dimensions is made, and the “superconvergence” phenomenon (4th order accuracy) is found for specific positioning of the source and target grids. Its limitation has also been studied. With rotated grids or grids of different sizes, this method can still reduce the error but fails to improve the order of the underlying interpolation technique. A series of numerical experiments on meshes with various translations, rotations, spacing ratios, and perturbations have been tested. Different interface treatments applied to cavity flow are compared. The 3D flow example shows that the BFECC method positively impacts the convergence of the domain decomposed problem. [ABSTRACT FROM AUTHOR]

In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the ϑ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence. [ABSTRACT FROM AUTHOR]

Multigrid methods, algebraic or geometric, commonly suffer from high frequency residuals after prolongation. This paper develops a stable approach to remove high frequency residuals for geometric multigrid methods for solving nonlinear advection–diffusion problems with degenerate coefficients. Here, a local problem is treated by optimization on subdomains with mesh refinements. Newton's method is utilized in the procedure and the iteration is completed when the residual in the subdomain is reduced to the given magnitude, usually set to be the average of residuals in the non-high-frequency domains. An oversampling technique is employed to further improve the stability by providing a definite flow path in regions where coefficients have high contrast and complex structures. Removing high frequency residuals before continuing the global Newton iteration improves global convergence behavior. [ABSTRACT FROM AUTHOR]

In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond2(V)) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V− 1, by which we prove that Cond 2 (V) = O (n 2) . This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores. [ABSTRACT FROM AUTHOR]

In this paper, the well-known shifted Grünwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysis and error estimates of the scheme are carried out in the maximum norm. An improved algorithm is presented based on the extrapolation technique to obtain high-order accuracy for the problems with low regularity. Various numerical experiments are given to support the theoretical finding and verify the effectiveness of the improved algorithm. It is illustrated that, by using the proposed algorithm, both accuracy and convergence rate can be significantly improved for solving the two-dimensional FBVPs as well as fractional diffusion equations with non-smooth solution. Especially, the convergence rate of corrected numerical solutions even can reach second-order in the case of solving problems involved one-sided fractional derivatives. [ABSTRACT FROM AUTHOR]

We construct a new Absorbing Boundary Condition (ABC) adapted to solving the Helmholtz equation in polygonal domains in dimension two. Quasi-continuity relations are obtained at the corners of the polygonal boundary. This ABC is then used in the context of domain decomposition where various stable algorithms are constructed and analysed. Next, the operator of this ABC is adapted to obtain a transmission operator for the Domain Decomposition Method (DDM) that is well suited for broken line interfaces. For each algorithm, we show the decrease of an adapted quadratic pseudo-energy written on the skeleton of the mesh decomposition, which establishes the stability of these methods. Implementation within a finite element solver (GMSH/GetDP) and numerical tests illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose an overlapping domain decomposition method (DDM) to improve the method of fundamental solution (MFS) for the elliptic partial differential equations. The MFS often solves the Poisson-type equations by the use of a particular solution which is obtained by the radial basis functions (RBFs) interpolation. Although the MFS is a boundary-type method, the interior points are essentially needed in approximating the particular solution. Consequently, when dealing with large-scale problems, the condition number of the RBF interpolation matrix considerably increases. To treat the ill-conditioning, we apply an overlapping DDM to localize the globally supported RBFs. In addition, we present a robust formulation of the Schwarz method based on MFS and the particular solution matrix. Furthermore, we show that the iteration involved in the Schwarz method is equivalent to the iterative Gauss–Seidel method. Some numerical examples will be given to demonstrate the performance of the suggested method. [ABSTRACT FROM AUTHOR]

We consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling. [ABSTRACT FROM AUTHOR]

The shallow water equation based on the generalized equal width (GEW) equation is a PDE that can be classified in the category of hyperbolic PDEs. The current paper concerns developing an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width model. For this aim, first, the space derivative is approximated by the local collocation method via a compact integrated radial basis function. Second, the time derivative is approximated by the method of lines that yields a system of nonlinear ODEs. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. Also, an adaptive collocation method based upon IRBF is presented for the solution of GEW. The effectiveness of our adaptive technique is shown in four examples such as interpolation, the motion of single solitary, two solitary waves and Maxwellian initial condition. It is explained in detail how the adaptive method refines the resolution. [ABSTRACT FROM AUTHOR]

In the paper, the space second-order conservative characteristic finite difference method based on essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) interpolation for solving two-dimensional conservative convection–diffusion equations in divergence form is developed. Combining the splitting technique, characteristic difference and mass correction method, a two-dimensional convection–diffusion equations are changed into the two-dimensional parabolic equations, where the convection term and unsteady term are considered as one term. The solutions and fluxes on the staggered meshes are computed by the splitting implicit solution-flux coupled scheme. Numerical experiments are presented to illustrate mass conservation and convergence. [ABSTRACT FROM AUTHOR]

Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

Computer simulations are used routinely in the automobile industry. The simulation algorithms and software are well developed for computations that are carried out on a small cluster of computers. To further reduce the design time and increase the fidelity of the computation, new algorithms and software are needed on larger computing systems with more processor cores. In this paper, we study a Newton–Krylov–Schwarz algorithm based 3D incompressible Navier–Stokes solver for solving unsteady, high Reynolds number flows around a car with realistic geometry and parameters. We also investigate the impact of the yaw angles and crosswind on the flow field. The numerical experiments show that the approach scales well with more than 8000 processor cores. [ABSTRACT FROM AUTHOR]