High-order Accurate Solution Research Articles

Laplace transforms for evaluation of Volterra integral equation of the first kind with highly oscillatory kernel

This paper focuses on the numerical solution for Volterra integral equations of the first kind with highly oscillatory Bessel kernel and highly oscillatory triangle function on the right-hand side. We first establish a new existence theorem of solutions for such equations, and then, the explicit formulas of the solution are derived based on Laplace and inverse Laplace transforms. Furthermore, high-order accurate numerical solutions for approximating the explicit solution are further deduced by applying the Clenshaw–Curtis–Filon method and other effective numerical methods. Preliminary numerical results not only show the exact formulas of the solution, but also present the accuracy of the approximations.

Mimetic discretization of the Eikonal equation with Soner boundary conditions

Motivated by a specific application in seismic reflection, the goal of this paper is to present a modified version of the Castillo–Grone mimetic gradient operators that allows for a high-order accurate solution of the Eikonal equation with Soner boundary conditions. The modified gradient operators utilize a non-staggered grid. In dimensions other than 1D, the modified gradient operators are expressed as Kronecker products of their corresponding 1D versions and some identity matrices. It is shown, that these modified 1D gradient operators are as accurate as the original gradient operators in terms of approximating first-order partial derivatives. It turns out, that in 1D one requires to solve two linear systems for finding a numerical solution of the Eikonal equation. Some examples show that the solution obtained by utilizing the modified operators increases its accuracy when incrementing the order of their approximation, something that does not occur when using the original operators. An iterative scheme is presented for the nonlinear 2D case. The method is of a quasi-Newton-like nature. At each iteration a linear system is built, with progressively higher-order stencils. The solution of the Fast Marching method is the initial guess. Numerical evidence indicates that high-order accurate solutions can be achieved.

Efficient Interpolation Strategies in Multiscale Multigrid Computation

We introduce two new interpolation strategies, SOR strategy and rotated grid strategy, to compute the fine grid high order accurate solution in multiscale multigrid computation based on the Richardson extrapolation technique for solving partial differential equations. These new interpolation strategies effectively accelerate or eliminate the iterative refinement process previously employed in multiscale multigrid computation to obtain high order accurate solution on the fine grid. Experimental results show that the proposed new interpolation strategies are much more efficient and faster than the previously used iterative refinement strategy to compute high order accurate solution on the fine grid.

Discontinuous Galerkin algorithms for fully kinetic plasmas

We present a new algorithm for the discretization of the non-relativistic Vlasov–Maxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strong-stability preserving Runge–Kutta method. Since the Vlasov equation in the Vlasov–Maxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution, including the use of a reduced high-order basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features.

High order accurate dual-phase-lag numerical model for microscopic heating in multiple domains

In this article, a characteristic-based dual-phase-lag numerical model based on finite difference method has been developed to predict the microscopic heating response in time as well as consideration of the micro-structured effect. High-order TVD (Total Variation Diminishing) schemes being oscillation-free can yield high-order accurate solutions without introducing wiggles and therefore are utilised in this work. A multi-domain approach integrated within the dual-phase-lag numerical model allows the computation of microscopic conjugate heat transfer problems. Effects of different phase-lag values on the behaviour of heat transfer are investigated. The model is capable of predicting temperature patterns transiting from the wave nature of heat propagation to additional diffusion being experienced within different solid regions via phonon–electron interaction or phonon scattering.

Assessment of a high-order discontinuous Galerkin method for internal flow problems. Part I: Benchmark results for quasi-1D, 2D waves propagation and axisymmetric turbulent flows

In this work we apply a high-order discontinuous Galerkin (DG) finite element method to inviscid and turbulent internal flow problems. The equations here considered are the quasi-1D, 2D Euler equations and the RANS and k−ω equations in axisymmetric coordinates. The method here proposed is designed to ensure high-order accuracy in ducts and engine-like geometries using both explicit and implicit schemes for the temporal discretization of the governing equations. Absorbing Sponge Layer (ASL) boundary conditions are implemented to minimize the reflection of out-going waves at open boundaries. A shock-capturing technique is used to control the oscillations of high-order solutions around shocks. Accurate solutions of the hyperbolic equations are performed by means of the five-stage fourth-order accurate Strong Stability Preserving Runge-Kutta scheme, while the implicit Backward-Euler scheme is adopted for efficient steady state simulations of internal turbulent flows. Two types of test-problems have been considered, one focusing on the potential of DG method to solve ideal quasi-1D and 2D waves propagation and shock phenomena that may occur in ducts, and the other on its feasibility to provide high-order accurate solutions of multi-dimensional internal turbulent flows in geometries typical of internal combustion engine (ICE) applications. To clearly illustrate the performance of the high-order DG method, the results are compared with exact solutions, experimental data and second-order accurate solutions obtained with a finite volume commercial code.

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

The numerical simulation of meso-, convective-, and microscale atmospheric flows requires the solution of the Euler or the Navier–Stokes equations. Nonhydrostatic weather prediction algorithms often solve the equations in terms of derived quantities such as Exner pressure and potential temperature (and are thus not conservative) and/or as perturbations to the hydrostatically balanced equilibrium state. This paper presents a well-balanced, conservative finite difference formulation for the Euler equations with a gravitational source term, where the governing equations are solved as conservation laws for mass, momentum, and energy. Preservation of the hydrostatic balance to machine precision by the discretized equations is essential because atmospheric phenomena are often small perturbations to this balance. The proposed algorithm uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization that yields high-order accurate solutions for smooth flows and is essentially nonoscillatory across strong gradients; however, the well-balanced formulation may be used with other conservative finite difference methods. The performance of the algorithm is demonstrated on test problems as well as benchmark atmospheric flow problems, and the results are verified with those in the literature.

Hybrid WENO schemes for polydisperse sedimentation models

Polydisperse sedimentation models can be described by a strongly coupled system of conservation laws for the concentration of each species of solids. Typical solutions for the sedimentation model considered for batch settling in a column include stationary kinematic shocks separating layers of sediment of different composition. This phenomenon, known as segregation of species, is a specially demanding task for numerical simulation due to the need of accurate numerical simulations. Very high-order accurate solutions can be constructed by incorporating characteristic information, available due to the hyperbolicity analysis made in Donat and Mulet [A secular equation for the Jacobian matrix of certain multispecies kinematic flow models, Numer. Methods Partial Differential Equations 26 (2010), pp. 159–175.] But characteristic-based schemes, see Bürger et al. [On the implementation of WENO schemes for a class of polydisperse sedimentation models, J. Comput. Phys. 230 (2011), pp. 2322–2344], are very expensive in terms of computational time, since characteristic information is not readily available, and they are not really necessary in constant areas, where a less complex method can obtain similar results. With this idea in mind, in this paper we develop a hybrid finite difference WENO scheme that only uses the characteristic information of the Jacobian matrix of the system in those regions where singularities exist or are starting to develop, while it uses a component-wise approximation of the scheme in smooth regions. We perform some experiments showing the computational gains that can be achieved by this strategy.

A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.

Hybrid spectral difference/embedded finite volume method for conservation laws

Recently, interests have been increasing towards applying the high-order methods to various engineering applications with complex geometries [30]. As a result, a family of discontinuous high-order methods, such as Discontinuous Galerkin (DG), Spectral Volume (SV) and Spectral Difference (SD) methods, is under active development. These methods provide high-order accurate solutions and are highly parallelizable due to the local solution reconstruction within each element. But, these methods suffer from the Gibbs phenomena when discontinuities are present in the flow fields. Various types of limiters [43–45] and artificial viscosity [46,48] have been employed to overcome this problem.A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method [56]. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field.The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.

A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of Open image in new window, where N + 1 and N0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.

Application of the RKDG method for gas dynamics problems

The Runge-Kutta discontinuous Galerkin method (RKDG method) is widely used nowadays. The RKDG method is characterized by a high-order accurate solution. As is well known, for ensuring the monotonic solution by this method, it is necessary to use slope limiters. In particular, it is necessary when the solution contains strong discontinuities. However, the application of limiters can negatively affect the accuracy of the solution. In this paper a way of saving the order of accuracy of the solution and ensuring its monotony is investigated.

The efficient implementation of correction procedure via reconstruction with graphics processing unit computing

Computational fluid dynamics (CFD) has long been a useful tool to model fluid flow problems across many engineering disciplines, and while problem size, complexity, and difficulty continue to expand, the demands for robustness and accuracy grow. Furthermore, generating high-order accurate solutions has escalated the required computational resources, and as problems continue to increase in complexity, so will computational needs such as memory requirements and calculation time for accurate flow field prediction. To improve upon computational time, vast amounts of computational power and resources are employed, but even over dozens to hundreds of central processing units (CPUs), the required computational time to compute solutions can be weeks, months, or longer, which is particularly true when generating high-order accurate solutions over large computational domains. One response to lower the computational time for CFD problems is to implement current CFD solvers on graphical processing units (GPUs). GPUs have illustrated the ability to solve problems orders of magnitude faster than their CPU counterparts with identical accuracy. The goal of the presented work is to combine a CFD solver and GPU computing with the intent to solve complex problems at a high-order of accuracy while lowering the computational time required to generate the solution. The CFD solver should have high-order spacial capabilities to evaluate small fluctuations and fluid structures not generally captured by lower-order methods (2nd and 1st order) and be efficient for the GPU architecture. This research combines the high-order Correction Procedure via Reconstruction (CPR) method for explicit time-stepping with compute unified device architecture (CUDA) from NVIDIA to reach these goals. In addition, the study demonstrates accuracy of the developed solver by comparing results with other solvers and exact solutions. Finally, to illustrate speed-ups, a single core Intel Xeon X5650 at 2.67GHz is compared against a Tesla C2070 and a K20x, where a maximum speed-up of nearly 70x is demonstrated.

Closed form solution of electromagnetic wave diffraction problem in a homogeneous bi‐isotropic medium

In this study, the problem of wave scattering of an electromagnetic field in a homogeneous bi‐isotropic medium by a perfectly conducting strip is theoretically analyzed. The crux of the study is a rigorous construction of a closed form solution in the complex domain. A series solution of electromagnetic plane wave diffraction problem in terms of the eigenfunctions that happen to be the generalized Gamma functions is found. In the transformed domain, the scattered field is physically interpreted by computing the convergence history, and thereby, higher order accurate solution has been obtained in complex domain in closed form. Copyright © 2014 John Wiley & Sons, Ltd.

A Fast High Order Iterative Solver for the Electromagnetic Scattering by Open Cavities Filled with the Inhomogeneous Media

AbstractThe scattering of the open cavity filled with the inhomogeneous media is studied. The problem is discretized with a fourth order finite difference scheme and the immersed interface method, resulting in a linear system of equations with the high order accurate solutions in the whole computational domain. To solve the system of equations, we design an efficient iterative solver, which is based on the fast Fourier transformation, and provides an ideal preconditioner for Krylov subspace method. Numerical experiments demonstrate the capability of the proposed fast high order iterative solver.

An exact and asymptotic analysis of a diffraction problem

The present analysis deals with diffraction of acoustic waves by an oscillating strip focusing on the exact and concise formulation of a series solution in the complex domain. The complete solution is represented by a series, the eigenfunctions of which are generalized gamma functions. An exact expression of this special function, with argument being ‘integer +1/2’, is derived. The convergence analysis of the series solution in transformed domain is discussed graphically. Finally, the scattered and total acoustic fields are obtained by exact and asymptotic evaluations of inverse Fourier transforms. The significance of the present investigation is the derivation of a high order accurate solution in a convenient form.

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

Agglomeration based discontinuous Galerkin discretization of the Euler and Navier–Stokes equations

In this work we exploit the flexibility associated to discontinuous Galerkin methods to perform high-order discretizations of the Euler and Navier–Stokes equations on very general meshes obtained by means of agglomeration techniques. Agglomeration is here considered as an effective mean to decouple the geometry representation from the solution approximation being alternative to standard isoparametric or breakthrough isogeometric discretizations. The mesh elements can be composed of a set of standard sub-elements belonging to a fine low-order mesh whose cardinality can be freely chosen according to the domain discretization capabilities. The number of mesh sub-elements still have an impact on the cost of numerical integration. Since the agglomerated elements are arbitrarily shaped, physical space orthonormal basis functions are here considered as a key ingredient to build a suitable discrete dG space. As a result we are allowed to perform high-order discretizations on top of the set of (possibly) subparametric sub-cells composing an agglomerated element. Our approach is validated on challenging viscous and inviscid test cases. We demonstrate the use of low-order meshes as a starting point to obtain high-order accurate solutions on coarse meshes.

Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

Implicit methods for finite-volume schemes on unstructured grids typically rely on a matrix-free implementation of GMRES and an explicit first-order accurate Jacobian for preconditioning. Globalization is typically achieved using a global timestep or a CFL based local timestep. We show that robustness of the globalization can be improved by supplementing the pseudo-timestepping method commonly used with a line search method. The number of timesteps required for convergence can be reduced by using a timestep that scales with the local residual. We also show that it is possible to form the high-order Jacobian explicitly at a reasonable computational cost. This is demonstrated for cases using both limited and unlimited reconstruction. This exact Jacobian can be used for preconditioning and directly in the GMRES method. The benefits of improvements in preconditioning and the elimination of residual evaluations in the inner iterations of the matrix-free GMRES method are substantial. Computational results focus on second- and fourth-order accurate schemes with some results for the third-order scheme. Overall computational cost for the matrix-explicit method is lower than the matrix-free method for all cases. The fourth-order matrix-explicit scheme is a factor of 1.6–3 faster than the matrix-free scheme while requiring about 50–100% more memory.

High-order accurate solution of the incompressible Navier–Stokes equations on massively parallel computers

The emergence of “petascale” supercomputers requires us to develop today’s simulation codes for (incompressible) flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power. In that spirit, we present a massively parallel high-order Navier–Stokes solver for large incompressible flow problems in three dimensions. The governing equations are discretized with finite differences in space and a semi-implicit time integration scheme. This discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers. The iterative solver for the pressure uses a highly efficient commutation-based preconditioner which is robust with respect to grid stretching. The efficiency of the implementation is further enhanced by carefully setting the (adaptive) termination criteria for the different iterative solvers. The computational work is distributed to different processing units by a geometric data decomposition in all three dimensions. This decomposition scheme ensures a low communication overhead and excellent scaling capabilities. The discretization is thoroughly validated. First, we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow. Second, we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria. Third, Orr–Sommerfeld and Squire eigenmodes for plane Poiseuille flow are simulated and compared to analytical results. Fourth, the practical applicability of the implementation is tested for transitional and turbulent channel flow. The results are compared to solutions from a pseudospectral solver. Subsequently, the performance of the commutation-based preconditioner for the pressure iteration is demonstrated. Finally, the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to O ( 10 4 ) processing units and O ( 10 11 ) grid points.