Strong Stability Preserving Runge Kutta Scheme Research Articles

An entropy conserving/stable discontinuous Galerkin solver in entropy variables based on the direct enforcement of entropy balance

Numerical schemes that guarantee the preservation of flow properties as prescribed by physical laws, ensure high-fidelity results and improved computational robustness. This work presents a highly accurate entropy conserving/stable solver for the solution of the compressible Euler equations. The method uses a modal Discontinuous Galerkin approximation and takes advantage of both entropy variables and the Direct Enforcement of Entropy Balance (DEEB), proposed by Abgrall [1] to fulfill the second law of thermodynamics at the discrete level. Thanks to the use of an orthonormal modal basis, the cost of adding the DEEB correction to the spatially discretized equations is negligible. Entropy variables are used directly, as unknowns, or to assemble the discrete operators while solving for the conservative variables. In the latter case, the conservative variables are “converted” into the entropy ones through an L2 “entropy projection”. If this strategy is coupled with DEEB, a very efficient alternative to the direct use of the entropy variables is obtained, with a dramatic improvement in robustness over the direct use of the conservative variables. For the time integration both an explicit Strong Stability Preserving Runge-Kutta scheme and the implicit Generalized Crank-Nicolson method are considered. The latter allows for the construction of a discrete entropy conservative/stable scheme both in space and time to be used as a reference. Convergence and conservation properties are demonstrated together with robustness and computational efficiency on a suite of two-dimensional test cases, showing that by coupling DEEB with entropy projection the solver robustness is improved while not affecting the stability limit.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

A meshless quasi-interpolation method for solving hyperbolic conservation laws based on the essenstially non-oscillatory reconstruction

In this paper, a quasi-interpolation method is proposed for solving hyperbolic conservation laws based on the essentially non-oscillatory (ENO) scheme. The hyperbolic equation is discretized in space with the finite difference ENO method and then the semi-discrete system is integrated by the strong stability preserving Runge-Kutta scheme. However, in the step of ENO reconstruction, the finite difference method is replaced by several quasi-interpolation schemes, including the multiquadric quasi-interpolation, the integral-type multiquadric quasi-interpolation and the cubic B-spline quasi-interpolation. Our quasi-interpolation method is simple and easy to implement since it doesn't need to solve any linear system. Moreover, it is also suitable for nonuniform grids and noisy sampling data. Nonlinear hyperbolic problems we target include one-dimensional and two-dimensional Burgers' equations and also the Euler equation. Numerical results demonstrate that the proposed quasi-interpolation ENO method is stable and has good accuracy.

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space and an explicit strong stability preserving Runge-Kutta scheme in time. When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues. The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way. For piecewise constant polynomials in space and explicit Euler in time, the stabilized scheme is monotone for scalar problems. For higher polynomial degrees p, our numerical experiments show convergence orders of p+1 for smooth flow and robust behavior in the presence of shocks.

An algorithm for finding optimal sensor placement and model uncertainty in data assimilation applied to Riemann problems

The sensor location and selection of model noise parameters play an utmost important role in the performance of data assimilation methods. Suboptimal sensor placement may lead to incomplete information and incorrectly chosen noise parameters can cause filter divergence. Till now, these issues were addressed separately, but in this work, we consider both. We present an algorithm that finds the optimal sensor location and non-Gaussian model noise parameters not only separately but also simultaneously. The optimal location was determined for a fixed number of sensors and the root mean square error was used to assess the accuracy of the data assimilation scheme. A direct search method was implemented and, due to the stochastic nature of the filtering method, we used a sample average approximation approach. To prove the usefulness of the algorithm, we selected Riemann problems in one and two dimensions, where the solutions contain shock waves, rarefaction waves, and contact discontinuities. A high order finite volume weighted essentially non-oscillatory scheme was used for the numerical approximation. We used the Rusanov Riemann solver for the numerical flux and the three-stage third-order strong stability preserving Runge-Kutta scheme for the time advancing. Numerical experiments showed that optimizing both sensor location and model noise parameters gives more accurate results without suffering from filter divergence. Uniform distribution of the sensors and manually setting the noise parameters caused the filter to fail to track the truth, especially near the discontinuities.

Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations

This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut (2004) [9] and Coquel and Perthame (1998) [22]. Then, we consider the DGSEM based on collocation of quadrature and interpolation points which relies on the framework introduced by Fisher and Carpenter (2013) [28] and Gassner (2013) [31]. We replace the physical fluxes in the integrals over discretization elements by entropy conservative numerical fluxes [Tadmor (1987) [71]], while ES numerical fluxes are used at element interfaces. We thus derive a two-point numerical flux satisfying the Tadmor's entropy conservation condition and use the numerical flux from the three-point scheme as ES flux. Time discretization is performed with a strong-stability preserving Runge-Kutta scheme. We then derive conditions on the numerical parameters to guaranty a semi-discrete entropy inequality as well as positivity of the cell average of the partial densities and internal energy of the fully discrete DGSEM at any approximation order. The later results allow to use existing limiters in order to restore positivity of nodal values within elements. The scheme also resolves exactly stationary material interfaces. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and high resolution of the scheme.

Analysis of RLW and MRLW equation using an improvised collocation technique with SSP-RK43 scheme

In this work, the regularized and the modified regularized long wave equations are solved using an improvised cubic B-spline collocation technique. Posteriori corrections are made in the cubic B-spline interpolant by forcing it to satisfy some specific interpolatory and end conditions, which results in the formation of improvised B-spline collocation technique. For temporal domain discretization, strong stability preserving Runge–Kutta scheme of third-order and four stages (SSP-RK43) is used. Seven important problems of the regularized and the modified regularized long wave equations are solved to demonstrate the applicability of the proposed technique. The motion of a single solitary wave, the interaction of two or more solitary waves, and the undulations of waves are shown graphically. These equations possess three invariants of motion namely mass, momentum, and energy, which need to remain constant with run time. For each problem, these invariants are calculated and it is observed that they coincide with exact values and remain preserved with time. The L2 and L∞ error norms are calculated at different time levels and CPU time is noted. Also, the stability analysis of the technique is carried out and is found to be conditionally stable.

An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato (1986) [5]) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [47]). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al. (2014) [10]; Gassner et al. (2016) [32]). Here, we build upon the framework provided in (Renac (2019) [55]) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (2013) [12], while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

A comparative analysis of explicit, IMEX and implicit strong stability preserving Runge-Kutta schemes

Selecting between explicit and implicit time marching schemes, as well as non-iterative explicit-implicit ones, is well understood on a linear stability basis. Implicit schemes are favored when either searching for steady-states or performing time accurate simulations of stiff equations due to their larger time steps. When nonlinear stability properties must be preserved, however, solution monotonicity is guaranteed by a time step restriction that puts into question the usefulness of implicit schemes. Five test cases are used in this paper for a comparative efficiency analysis of explicit, implicit and implicit-explicit Strong Stability Preserving (SSP) Runge-Kutta (RK) schemes. They indicate that explicit SSP-RK schemes are, in general, the most efficient ones. It is important to emphasize that great care must be exercised when generalizing these conclusions, since such an analysis is inherently software, hardware, numerics and model dependent. Nevertheless, this study provides the first systematic evidence in the literature confirming these trends, to the best of our knowledge. Finally, implicit-explicit SSP-RK schemes do perform well, even though they are not optimal. Hence, there is a need for the development of their optimal versions before any conclusions can be drawn about their efficiency.

High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics

This paper studies high-order accurate entropy stable nodal discontinuous Galerkin (DG) schemes for the ideal special relativistic magnetohydrodynamics (RMHD). It is built on the modified RMHD equations with a particular source term, which is analogous to the Powell's eight-wave formulation and can be symmetrized so that an “entropy pair” is obtained. We design an affordable “fully consistent” two-point entropy conservative flux, which is not only consistent with the physical flux, but also maintains the zero parallel magnetic component, and then construct high-order accurate semi-discrete entropy stable DG schemes based on the quadrature rules and the entropy conservative and stable fluxes. The Lax-Friedrichs flux is proven to be entropy stable when its viscosity coefficient is chosen as the speed of light, and the other two entropy stable fluxes are also derived by adding the dissipation terms to our proposed entropy conservative flux and compared to the Lax-Friedrichs flux by using numerical experiments in terms of the CPU time and accuracy. All three entropy stable numerical fluxes maintain the zero parallel magnetic component in one dimension. The resulting entropy stable DG schemes satisfy the semi-discrete “entropy inequality” for the given “entropy pair” and are integrated in time by using the high-order explicit strong stability preserving Runge-Kutta schemes to get further the fully-discrete nodal DG schemes. Extensive numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our schemes with the help of the TVB limiter. Moreover, our entropy conservative flux is compared to an existing flux through some numerical tests. The results show that the zero parallel magnetic component in the numerical flux can help to decrease the error in the parallel magnetic component in one-dimensional tests, but two entropy conservative fluxes give similar results since the error in the magnetic field divergence seems dominated in the two-dimensional tests.

Positivity-Preserving Finite Difference WENO Scheme for Ten-Moment Equations with Source Term

We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Runge–Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on the time step. The proposed method can be used with any WENO reconstruction scheme and we demonstrate it with fifth order accurate WENO-JS, WENO-Z and WENO-AO schemes. An adaptive CFL strategy is developed which can be more efficient than the use of reduced CFL for positivity preservation. Numerical results show that high order accuracy and positivity preservation are achieved on a range of test problems.

Optimal stabilization and time step constraints for the forward Euler-Local Discontinuous Galerkin method applied to fractional diffusion equations

A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order 1<α<2 is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order h1−α and using a von Neumann analysis, stability conditions proportional to hα are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory.

Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows

In this work, we consider the discretization of nonlinear hyperbolic systems in nonconservative form with the high-order discontinuous Galerkin spectral element method (DGSEM) based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [39]). We present a general framework for the design of such schemes that satisfy a semi-discrete entropy inequality for a given convex entropy function at any approximation order. The framework is closely related to the one introduced for conservation laws by Fisher and Carpenter (2013) [27] and Gassner (2013) [32] and relies on the modification of the integral over discretization elements where we replace the physical fluxes by entropy conservative numerical fluxes from Castro et al. (2013) [12], while entropy stable numerical fluxes are used at element interfaces. Time discretization is performed with strong-stability preserving Runge–Kutta schemes. The present method is first validated on the Burgers and Euler equations in nonconservative form. We then use this framework for the discretization of two-phase flow systems in one space-dimension: a 2×2 system with a nonconservative product associated to a linearly-degenerate field for which the DGSEM fails to capture the physically relevant solution, and the isentropic Baer–Nunziato model. For the latter, we derive conditions on the numerical parameters of the discrete scheme to further keep positivity of the partial densities and a maximum principle on the void fractions. Numerical experiments support the conclusions of the present analysis and highlight stability and robustness of the present schemes.

An improved non-linear weights for seventh-order weighted essentially non-oscillatory scheme

In this article, the construction and implementation of a seventh order weighted essentially non-oscillatory scheme is reported for solving hyperbolic conservation laws. Local smoothness indicators are constructed based on L1-norm, where a higher order interpolation polynomial is used with each derivative being approximated to the fourth order of accuracy with respect to the evaluation point. The global smoothness indicator so constructed ensures the scheme achieves the desired order of accuracy. The scheme is reviewed in the presence of critical points and verified the numerical accuracy, convergence with the help of linear scalar test cases. Further, the scheme is implemented to non-linear scalar and system of equations in one and two dimensions. As the formulation is based on method of lines, to move forward in time linear strong-stability-preserving Runge–Kutta scheme (lSSPRK) for the linear problems and the fourth order nonlinear version of five stage strong stability preserving Runge–Kutta scheme (SSPRK(5, 4)) for nonlinear problems is used.

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.

Quasi One-Dimensional Model of Natural Draft Wet-Cooling Tower Flow, Heat and Mass Transfer

The article deals with the development of CFD (Computational Fluid Dynamics) model of natural draft wet-cooling tower flow, heat and mass transfer. The moist air flow is described by the system of conservation laws along with additional equations. Moist air is assumed to be homogeneous mixture of dry air and water vapour. Liquid phase in the fill zone is described by the system of ordinary differential equations. Boundary value problem for the system of conservation laws is discretized in space using Kurganov-Tadmor central scheme and in time using strong stability preserving Runge-Kutta scheme. Initial value problems in the fill zone is solved by using standard fourth order Runge-Kutta scheme. The interaction between liquid water and moist air is done by source terms in governing equations.

Numerical simulation of two dimensional sine-Gordon solitons using modified cubic B-spline differential quadrature method

In this paper, a modified cubic B-spline differential quadrature method (MCB-DQM) is employed for the numerical simulation of two-space dimensional nonlinear sine-Gordon equation with appropriate initial and boundary conditions. The modified cubic B-spline works as a basis function in the differential quadrature method to compute the weighting coefficients. Accordingly, two dimensional sine-Gordon equation is transformed into a system of second order ordinary differential equations (ODEs). The resultant system of ODEs is solved by employing an optimal five stage and fourth-order strong stability preserving Runge–Kutta scheme (SSP-RK54). Numerical simulation is discussed for both damped and undamped cases. Computational results are found to be in good agreement with the exact solution and other numerical results available in the literature.

Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method

In this paper, a numerical solution of two dimensional nonlinear coupled viscous Burger equation is discussed with appropriate initial and boundary conditions using the modified cubic B-spline differential quadrature method. In this method, the weighting coefficients are computed using the modified cubic B-spline as a basis function in the differential quadrature method. Thus, the coupled Burger equation is reduced into a system of ordinary differential equations. An optimal five stage and fourth-order strong stability preserving Runge–Kutta scheme is applied for solving the resulting system of ordinary differential equations. The accuracy of the scheme is illustrated by taking two numerical examples. Computed results are compared with the exact solutions and other results available in literature. Obtained numerical result shows that the described method is efficient and reliable scheme for solving two dimensional coupled viscous Burger equation.

Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations

In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier–Stokes equations.Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation.The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge–Kutta scheme, SSPRK4.

Application of compact finite-difference schemes to simulations of stably stratified fluid flows

This paper presents a comparison of the results of numerical simulations obtained by two different numerical methods for one specific case of stably stratified incompressible flow. The focus in this paper is on the numerical results obtained using some of the compact finite-difference discretizations introduced in the paper [1]. The numerical scheme itself follows the principle of semi-discretisation, with high order compact discretisation in space, while the time integration is carried out by the Strong Stability Preserving Runge–Kutta scheme. Results are compared against the reference solution obtained by the AUSM finite volume method. The test case used to demonstrate the capabilities of selected numerical methods represents the flow of stably stratified fluid over low, smooth, hill-like wall mounted obstacle.