Order Spatial Accuracy Research Articles

High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals. Aiming at the one-dimensional time-fractional telegraph equation, a class of explicit-implicit (E-I) difference methods and implicit-explicit (I-E) difference methods are proposed. The two methods are based on a combination of the classical implicit difference method and the classical explicit difference method. Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, \begin{document}$ 2-\alpha $\end{document} order time accuracy, and have significant time-saving, their calculation efficiency is higher than the classical implicit scheme. The research shows that the E-I and I-E difference methods constructed in this paper are effective for solving the time-fractional telegraph equation.

ChromaTech: A discontinuous Galerkin spectral element simulator for preparative liquid chromatography

ChromaTech is a simulator for preparative liquid chromatography processes with mass transport described by the pore and surface diffusion general rate model. A discontinuous Galerkin spectral element method is used for spatial discretization with exponential decay of approximation errors within elements. The code is validated by numerically reproducing a high-precision reference obtained with CADET-semi-analytic. The performance of ChromaTech is tested by comparing against CADET, a dedicated code based on a weighted essentially non-oscillatory finite volume method with second (low) order spatial accuracy. Reassuringly, ChromaTech provides exactly the same chromatograms as CADET for multicomponent protein purification cases with linear and non-linear adsorption dynamics. However, the numerical results show, that ChromaTech has superior efficiency in terms of computational cost and discrete problem size without compromising stability. The spatial discretization is the major difference between the two codes for solution of the pore and surface diffusion general rate model. Thus, it demonstrates, that spectral methods are not just competitive with second (low) order accurate methods often used by default, but simply a superior approach for spatial discretization of liquid chromatography flow problems in terms of computational efficiency.

Maximum norm error estimates of a linear CADI method for the Klein–Gordon–Schrödinger equations

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrödinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method. Finally, some numerical experiments are presented to support the theoretical results.

Higher Order Compact Schemes for the Spatial Discretization of Linear Parabolic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative of Linear Parabolic Partial Differential Equations (LPPDE), on the non-boundary nodes. Euler time integration for the temporal derivative, the Crank-Nicholson (CN)scheme for the spatial derivatives and the source term are used. The higher order spatial accuracy of the developed, central difference based compact schemes for the one and two-dimensional diffusion equations are validated, by solving numerically two test problems. The compact scheme based calculations involve a tridiagonal matrix vector multiplication and a vector vector subtraction. The interest is to demonstrate the higher order spatial accuracy and the better rate of convergence of the solution produced using the developed compact 4th order scheme, when compared with the same produced, using the conventional 2nd order scheme. The assessment is made, in terms of the discrete or norm of the true error of the converged numerical solution.

Finite Difference Investigation of a Polluted Non-Isothermal Variable-Viscosity Porous Media Flow

We extend previous studies of channel flows to porous media flows with combined effects ofboth heat and mass transfer. We consider a temperaturedependent viscosity fluid and a concentrationdependent diffusivity in an unsteady and pressuredriven nonisothermal Brinkman flow. This leads to the governing equations for velocity, concentration and temperature. By lagging nonlinear coefficients, in time, a convergent finite difference scheme is formulated. We adopt the method of manufactured solutions to verify the convergence and second order spatial accuracy of the scheme. The impact of the flow parameters on the flow fields are numerically investigated. The results show that increase in the Darcy number and temperature parameter both increase the velocity while the increase in the pollutant diffusion parameter decreases the pollutant concentration.

Compact Schemes for the Spatial Discretization of Linear Elliptic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative 2 2 1 f x   and 2 2 2 f x   of Linear Elliptic Partial Differential Equations (LEPDE) ,on the non-boundary nodes, located along a particular horizontal grid line for 2 2 1 f x   and along a particular vertical grid line for 2 2 2 f x   of a two-dimensional structured Cartesian uniform gr.d. The aim of the numerical experiment is to demonstrate the higher order spatial accuracy and better rate of convergence of the solution, produced using the developed compact scheme. Further, these solutions are compared with the same, produced using the conventional 2nd order scheme. The comparison is made, in terms of the discrete l2 &l norms, of the true error. The true error is defined as, the difference between the computed numerical and the available exact solution, of the chosen test problems. It is computed on every non-boundary node bounded in the computational domain.

WENO schemes on unstructured meshes using a relaxed a posteriori MOOD limiting approach

In this paper a relaxed formulation of the a posteriori Multi-dimensional Optimal Order Detection (MOOD) limiting approach is introduced for weighted essentially non-oscillatory (WENO) finite volume schemes on unstructured meshes. The main goal is to minimise the computational footprint of the MOOD limiting approach by employing WENO schemes—by virtue of requiring a smaller number of cells to reduce their order of accuracy compared to an unlimited scheme. The key characteristic of the present relaxed MOOD formulation is that the Numerical Admissible Detector (NAD) is not uniquely defined for all orders of spatial accuracy, and it is relaxed when reaching a 2nd-order of accuracy. The augmented numerical schemes are applied to the 2D unsteady Euler equations for a multitude of test problems including the 2D vortex evolution, cylindrical explosion, double-Mach reflection, and an implosion. It is observed that in many events, the implemented MOOD paradigm manages to preserve symmetry of the forming structures in simulations, an improvement comparing to the non-MOOD limited counterparts which cannot be easily obtained due to the multi-dimensional reconstruction nature of the schemes.

The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

A new fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized accurately. The main idea is to create auxiliary unknowns at existing grid locations which increases the degrees of freedom of the initial problem. These auxiliary unknowns allow to impose various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

High Order Compact Generalized Finite Difference Methods for Solving Inviscid Compressible Flows

This paper presents a novel generalized finite difference method that can achieve arbitrary order of accuracy on a compact stencil nodal set. Accurate reconstruction and flux evaluation are two key steps to achieve high order spatial accuracy. A newly developed variational reconstruction approach is utilized to obtain the piecewise higher order polynomial distribution of flow variables. The implementation of boundary conditions is of critical importance and a flexible variational extrapolation technique is proposed for the high order boundary treatment. The numerical flux derivatives are evaluated using a simple and efficient hybrid approach, in which the linear and high order terms of the flux function are treated differently. Several test cases are solved to verify the accuracy, efficiency, and shock capturing capability of the proposed numerical schemes for inviscid compressible flows.

A High-Order Compact (HOC) Implicit Difference Scheme and a Multigrid Method for Solving 3D Unsteady Reaction Diffusion Equations

A high-order compact (HOC) implicit difference scheme is proposed for solving three-dimensional (3D) unsteady reaction diffusion equations. To discretize the spatial second-order derivatives, the fourth-order compact difference operators are used, and the third- and fourth-order derivative terms, which appear in the truncation error term, are also discretized by the compact difference method. For the temporal discretization, the multistep backward Euler formula is used to obtain the fourth-order accuracy, which matches the spatial accuracy order. To accelerate the traditional relaxation methods, a multigrid method is employed, and the computational efficiency is greatly improved. Numerical experiments are carried out to validate the accuracy and efficiency of the present method.

Novel numerical method of the fractional cable equation

In this article, mainly based on the second order compact approximation of first order derivative, the novel numerical method with second order temporal accuracy and fourth order spatial accuracy is proposed to solve the fractional cable equation. The numerical analysis involving convergence and stability of the novel numerical method subject to strict and detailed discussion. In addition, the numerical experiment strongly support the theoretical analysis results.

A new parallel difference algorithm based on improved alternating segment Crank\u2013Nicolson scheme for time fractional reaction\u2013diffusion equation

The fractional reaction–diffusion equation has profound physical and engineering background, and its rapid solution research is of important scientific significance and engineering application value. In this paper, we propose a parallel computing method of mixed difference scheme for time fractional reaction–diffusion equation and construct a class of improved alternating segment Crank–Nicolson (IASC–N) difference schemes. The class of parallel difference schemes constructed in this paper, based on the classical Crank–Nicolson (C–N) scheme and classical explicit and implicit schemes, combines with alternating segment techniques. We illustrate the unique existence, unconditional stability, and convergence of the parallel difference scheme solution theoretically. Numerical experiments verify the theoretical analysis, which shows that the IASC–N scheme has second order spatial accuracy and 2-alpha order temporal accuracy, and the computational efficiency is greatly improved compared with the implicit scheme and C–N scheme. The IASC–N scheme has ideal computation accuracy and obvious parallel computing properties, showing that the IASC–N parallel difference method is effective for solving time fractional reaction–diffusion equation.

High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes with WENO Adaptive-Order reconstruction on unstructured meshes

In this work we present a conservative WENO Adaptive Order (AO) reconstruction operator applied to an explicit one-step Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) method. The spatial order of accuracy is improved by reconstructing higher order piecewise polynomials of degree M>N, starting from the underlying polynomial solution of degree N provided by the DG scheme. High order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor that arises from a one-step time integration procedure. As a result, space-time polynomials of order M+1 are obtained and used to perform the time evolution of the numerical solution adopting a fully explicit DG scheme.To maintain algorithm simplicity, the mesh motion is restricted to be carried out using straight lines, hence the old mesh configuration at time tn is connected with the new one at time tn+1 via space-time segments, which result in space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our algorithm falls into the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where the governing PDE system is directly discretized relying on a space-time conservation formulation and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. A local rezoning strategy might be used in order to locally optimize the mesh quality and avoiding the generation of invalid elements with negative determinant. The proposed approach reduces to direct ALE finite volume schemes if N=0, while explicit direct ALE DG schemes are recovered in the case of N=M.In order to stabilize the DG solution, an a priori WENO based limiting technique is employed, that makes use of the numerical solution inside the element under consideration and its neighbor cells to find a less oscillatory polynomial approximation. By using a modal basis in a reference element, the evaluation of the oscillation indicators is very easily and efficiently carried out, hence allowing higher order modes to be properly limited, while leaving the zero-th order mode untouched for ensuring conservation.Numerical convergence rates for 2≤N,M≤4 are presented as well as a wide set of benchmark test problems for hydrodynamics on moving and fixed unstructured meshes.

Numerical experiments of flux difference splitting methods with high resolution scheme for supersonic flows

In this work, we have carried out the assessment of a high resolution scheme for unsteady compressible flow. For high order spatial accuracy, we have used fifth order weighted essentially non oscillatory (WENO) scheme. This scheme is applied to four flux difference splitting (FDS) methods: Harten-Lax-van-Leer (HLL), Roe solver, Harten-Lax-van Leer-Contact (HLLC), and Rusanov methods. We have compared results of these flux schemes with each other. WENO scheme is used for the reconstruction of left and right state variable across the cell interface for high resolution. The reconstruction procedure is performed in terms of primitive variables instead of conservative variable, in order to avoid spurious oscillation. We have considered two test cases: shock wave reflection and supersonic viscous flow over a flat plate, to access the performance of FDS schemes. An explicit third order TVD Runge-Kutta method is used for advancement of solution in time. The present results are compared with available numerical solutions. WENO-HLLC has good shock capturing capabilities as compare to WENO-Roe, WENO-HLL and WENO-Rusanov methods. It also provides best results inside and outside the boundary layer.

The Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation

In this paper, the Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation is studied. Most of the previous methods are low in temporal and spatial accuracy order. Based on the idea of Monte Carlo Markov chain method and compact finite difference schemes, a probability model for solving the modified anomalous fractional sub-diffusion equation is established. Numerical examples are given to show the feasibility of the proposed scheme. Compared with the compact finite difference method, the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. And it is also applied to solve partial differential equation in irregular domains.

Padé numerical schemes for the sine-Gordon equation

The sine-Gordon equation turn up in several problems in science and engineering. Although it is integrable, in practical applications, its numerical solution is powerful and versatile. Four novel implicit finite difference methods based on (q, s) Padé approximations with (q+s)th order in space have been developed and analyzed for this equation; all share the same treatment for the nonlinearity and integration in time. Concretely, (0,4), (2,2), (2,4), and (4,4) Padé methods; additionally, the energy conserving, Strauss–Vázquez scheme has been considered in a (0,2) Padé implementation. These methods have been compared among them for both the kink–antikink and breather solutions in terms of global error, computational cost and energy conservation. The (0,4) and (2,4) Padé methods are the most cost-effective ones for small and large global error, respectively. Our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations.

Two approaches to creating a turbulence model with increased temporal accuracy

When modeling a turbulent fluid flow, an Approximate Deconvolution Model (ADM) is sometimes chosen - in particular, due to the high order spatial accuracy. A method has been presented in [1], that demonstrates an approach to increase the temporal accuracy of the ADM, by combining it with the Deferred Correction method (DCM). The resulting model, DC-ADM, is at least second order accurate in both space and time, and theoretically an arbitrarily high order of accuracy is achievable, provided that enough computational resources are available. However, in some applications, especially when random input data are present, the efficiency of the ADM becomes at least as important, as its accuracy. In order to keep the second order accuracy, but reduce the computational time, we propose a modification of the DC-ADM: in the first, defect step, we replace the turbulence model with the computationally cheaper artificial viscosity approximation. Obviously, this defect step approximation loses accuracy, compared with that obtained in the DC-ADM method, but we will show that the accuracy is recovered in the correction step. The numerical tests demonstrate that the new approach is more efficient, if the compared algorithms are run sequentially: the results from the proposed method are obtained in 35% less time, than those obtained by DC-ADM. The tests also demonstrate that there is no loss of quality of the solutions, when switching from the DC-ADM to the new model. When parallelization is introduced, the methods are comparable in both accuracy and computational time.

Convergence of a high-order compact finite difference scheme for the Klein-Gordon-Schrödinger equations

In this paper, a conservative and linearly implicit finite difference scheme with second order temporal accuracy and eighth order spatial accuracy by means of the Richardson extrapolation method is proposed for approximating solution to the initial-boundary problem for the Klein-Gordon-Schrödinger equations. Furthermore, the difference solution is shown to be bounded by taking advantage of the fact that the proposed scheme possesses two conservation laws, and with the aid of the discrete energy method, the difference scheme is demonstrated to be convergent in the maximum norm. Finally, numerical experiments are assigned to support the theoretical results.

Parallel performance assessment of a pseudo-spectral solver for transition and turbulent boundary layer flows

The computational performance and accuracy of a parallel pseudo-spectral solver is assessed, and the solver is validated for direct numerical simulations (DNS) of bypass transition and turbulent boundary layer flows. The solver shows spatial order of accuracy of 6.8 and 3.7 for laminar and turbulent flow simulations, respectively, and scalability up to 16 thousand processors. DNS predictions for plane channel and flat plate bypass transition flows compare very well with the benchmark results available in the literature. Analysis of the turbulent structures shows counter-rotating longitudinal structures in the sub/buffer layers, which inducing lifting on each other to generate series of sweep and ejection events. These events generate shear stress which transfers energy from mean flow to streamwise turbulent fluctuations. The energy is then redistributed to other turbulent velocity fluctuations via pressure-strain. Temporal DNS of bypass pass transition flow compare very well with spatial DNS using an order of magnitude lower computational cost, thus is identified to be a viable alternative for transition flow physics simulations.