Strong Stability Preserving Runge-Kutta Methods Research Articles

Strong Stability Preserving General Linear Methods with Runge–Kutta Stability

We investigate strong stability preserving (SSP) general linear methods (GLMs) for systems of ordinary differential equations. Such methods are obtained by the solution of the minimization problems with nonlinear inequality constrains, corresponding to the SSP property of these methods, and equality constrains, corresponding to order and stage order conditions. These minimization problems were solved by the sequential quadratic programming algorithm implemented in MATLAB $$^{\circledR }$$ subroutine fmincon.m starting with many random guesses. Examples of transformed SSP GLMs of order $$p = 1, 2, 3$$ , and 4, and stage order $$q = p$$ have been determined, and suitable starting and finishing procedures have been constructed. The numerical experiments performed on a set of test problems have shown that transformed SSP GLMs constructed in this paper are more accurate than transformed SSP DIMSIMs and SSP Runge–Kutta methods of the same order.

Hybrid DG/FV schemes for magnetohydrodynamics and relativistic hydrodynamics

This paper presents a high order hybrid discontinuous Galerkin/finite volume scheme for solving the equations of the magnetohydrodynamics (MHD) and of the relativistic hydrodynamics (SRHD) on quadrilateral meshes. In this approach, for the spatial discretization, an arbitrary high order discontinuous Galerkin spectral element (DG) method is combined with a finite volume (FV) scheme in order to simulate complex flow problems involving strong shocks. Regarding the time discretization, a fourth order strong stability preserving Runge–Kutta method is used. In the proposed hybrid scheme, a shock indicator is computed at the beginning of each Runge–Kutta stage in order to flag those elements containing shock waves or discontinuities. Subsequently, the DG solution in these troubled elements and in the current time step is projected onto a subdomain composed of finite volume subcells. Right after, the DG operator is applied to those unflagged elements, which, in principle, are oscillation-free, meanwhile the troubled elements are evolved with a robust second/third order FV operator. With this approach we are able to numerically simulate very challenging problems in the context of MHD and SRHD in one, and two space dimensions and with very high order polynomials. We make convergence tests and show a comprehensive one- and two dimensional testbench for both equation systems, focusing in problems with strong shocks. The presented hybrid approach shows that numerical schemes of very high order of accuracy are able to simulate these complex flow problems in an efficient and robust manner.

Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. The search for high order strong stability preserving time-stepping methods with high order and large allowable time-step has been an active area of research. It is known that implicit SSP Runge–Kutta methods exist only up to sixth order; however, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge–Kutta methods of any linear order. In the current work we find implicit SSP Runge–Kutta methods with high linear order $$p_{lin} \le 9$$ and nonlinear orders $$p=2,3,4$$ , that are optimal in terms of allowable SSP time-step. Next, we formulate a novel optimization problem for implicit–explicit (IMEX) SSP Runge–Kutta methods and find optimized IMEX SSP Runge–Kutta pairs that have high linear order $$p_{lin} \le 7$$ and nonlinear orders up to $$p=4$$ . We also find implicit methods with large linear stability regions that pair with known explicit SSP Runge–Kutta methods. These methods are then tested on sample problems to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.

Numerical modeling of submarine turbidity currents over erodible beds using unstructured grids

Second-order central-upwind schemes proposed by Bryson et al. (2011) for the Saint–Venant system have two very attractive properties: well-balanced and positivity preserving, which are originally designed for constant fluid density and fixed beds in Bryson et al. (2011). For the turbidity current system with variable density over erodible beds, such desired properties can be obtained by developing a well-balanced and positivity preserving central-upwind scheme following the ideas in Bryson et al. (2011). To this end, in this paper, a coupled numerical model for two-dimensional depth-averaged turbidity current system over erodible beds is developed using finite volume method on triangular grids. The proposed numerical model is second-order accurate in space using piecewise linear reconstruction and third-order accurate in time using a strong stability preserving Runge–Kutta method. Applying the central-upwind method to estimate numerical fluxes through cell interfaces, the model can successfully deal with sharp gradients in turbidity flows. The developed numerical model can preserve the well-balanced property over irregular bottom, guarantee the non-negative turbidity current depth over erodible beds, and preserving the positivity of suspended sediment. These features of the developed numerical model and its robustness and accuracy are demonstrated in several numerical tests.

Dense Output for Strong Stability Preserving Runge–Kutta Methods

We investigate dense output formulae (also known as continuous extensions) for strong stability preserving (SSP) Runge---Kutta methods. We require that the dense output formula also possess the SSP property, ideally under the same step-size restriction as the method itself. A general recipe for first-order SSP dense output formulae for SSP methods is given, and second-order dense output formulae for several optimal SSP methods are developed. It is shown that SSP dense output formulae of order three and higher do not exist, and that in any method possessing a second-order SSP dense output, the coefficient matrix A has a zero row.

Numerical Simulations of a Polydisperse Sedimentation Model by Using Spectral WENO Method with Adaptive Multiresolution

In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11 species.

A novel approach for numerical computation of Burgers’ equation in (1 + 1) and (2 + 1) dimensions

This paper proposes a new scheme termed as modified extended cubic B-Spline differential quadrature (mECDQ) method for time dependent partial differential equations. Specially, the numerical computation of the Burgers’ equation is obtained using mECDQ method. First of all the modified extended cubic B-splines are used as a set of basis function in DQ to evaluate the weighting coefficients. The mECDQ method converts the initial boundary value system of Burgers’ equation into a initial value system of ordinary differential equations (ODEs). The resulting system is solved by using an optimal five stage four order strong stability preserving Runge–Kutta method (SSP-RK54). The accuracy and efficiency of the method is illustrated by considering five test problems. The proposed results are compared with the exact solutions in terms of L2 and L∞ error norms and the existing results. The mECDQ scheme produces better results than the results due to almost all the existing schemes. The stability analysis of the scheme is also carried out using the matrix stability analysis method for various grid points. This shows that mECDQ scheme is conditionally stable.

Void Fraction Based Two Phase Flow Model of Supersaturated Moist Air

The article deals with the developed of the model of supersaturated moist air flow with condensation. The proposed model is suitable especially for the cases where the rate of change of state variables in the flow field is slow like the flow in natural draft wet-cooling towers and in general the low Mach number flow. Void fraction of gas phase is included in governing equations. Homogeneous equilibrium model, where the two phases are well mixed and have the same velocity, is used. The results of the numerical solutions of the convergent nozzle are presented. The numerical fluxes are based on AUSM + -up scheme and temporal discretization is based on four stage order three explicit strong stability preserving Runge-Kutta method with non-negative coefficients.

Numerical solutions of two-dimensional unsteady convection–diffusion problems using modified bi-cubic B-spline finite elements

ABSTRACTThis research study presents a numerical scheme to compute approximate solutions of two dimensional unsteady convection–diffusion equation. We used collocation of modified bi-cubic B-spline functions for dependent variable u and for its derivatives w.r.t. space variables x and y. Strong stability preserving Runge–Kutta method (SSP-RK54) has been used for solving system of first-order ordinary differential equations obtained from the collocation form of the partial differential equation. We did not linearize the nonlinear terms by using any transformation or linearization method. The number of computations and the required storage space is very less for the proposed scheme. Four examples have been taken as described in available literature to demonstrate the effect and utility of the proposed scheme. These numerical experiments show that the obtained results are not only quite satisfactory w.r.t. the exact solutions but also competent with the solutions available in earlier research studies. Computational complexity of the proposed scheme has been discussed and shown that it is , where p is total number of nodes. The proposed scheme is easy to implement and the size of required computational work is very small. Moreover, using this scheme, we can compute approximate solutions not only at the mesh points but at any other point of the solution domain as well.

Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip+-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in Lp-norms (1 ≤ p < ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip+-stability property numerically and give some examples of when it holds strictly and when it is violated.

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order

High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability timestepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge–Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge–Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge–Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may be still be worthwhile to use methods with higher nonlinear order.

Strong Stability for Runge–Kutta Schemes on a Class of Nonlinear Problems

In this paper we consider Strong Stability Preserving (SSP) properties for explicit Runge---Kutta (RK) methods applied to a class of nonlinear ordinary differential equations. We define new modified threshold factors that allow us to prove properties, provided that they hold for explicit Euler steps. For many methods, the stepsize restrictions obtained are sharper than the ones obtained in terms of the Kraaijevanger's coefficient in the SSP theory. In particular, for the classical 4-stage fourth order method we get nontrivial stepsize restrictions. Furthermore, the order barrier $$p\le 4$$ p ≤ 4 for explicit SSP RK methods is not obtained. An open question is the existence of explicit RK schemes with order $$p\ge 5$$ p ? 5 and nontrivial modified threshold factor. The numerical experiments done illustrate the results obtained.

Strong Stability Preserving Explicit Runge--Kutta Methods of Maximal Effective Order

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods - like classical order five methods - require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge-Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice.

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers—Fisher equation. The method is based on the direct weak formulation of the Burgers—Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge—Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.

Surprising computations

In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We examine several instances of such computations. These include (i) solving Hamiltonian ODE systems using almost conservative explicit Runge–Kutta methods, (ii) applying splitting methods for the nonlinear Schrödinger equation, and (iii) applying strong stability preserving Runge–Kutta methods in conjunction with weighted essentially non-oscillatory semi-discretizations for nonlinear conservation laws with discontinuous solutions.For each problem and method class we present a simple numerical example that yields results that in our experience many active researchers are finding unexpected and unintuitive. Each numerical example is then followed by an explanation and a resolution of the practical problem.

A Simplified h-box Method for Embedded Boundary Grids

We present a simplified $h$-box method for integrating time-dependent conservation laws on embedded boundary grids using an explicit finite volume scheme. By using a method of lines approach with a strong stability preserving Runge-Kutta method in time, the complexity of our previously introduced $h$-box method is greatly reduced. A stable, accurate, and conservative approximation is obtained by constructing a finite volume method where the numerical fluxes satisfy a certain cancellation property. For a model problem in one space dimension using appropriate limiting strategies, the resulting method is shown to be total variation diminishing. In two space dimensions, stability is maintained by using rotated $h$-boxes as introduced in previous work [M. J. Berger and R. J. LeVeque, Comput. Systems Engrg., 1 (1990), pp. 305-311; C. Helzel, M. J. Berger, and R. J. LeVeque, SIAM J. Sci. Comput., 26 (2005), pp. 785-809], but in the new formulation, $h$-box gradients are taken solely from the underlying Cartesian grid, which also reduces the computational cost.

Finite Volume Model for Two-Dimensional Shallow Environmental Flow

This paper presents the development of a two-dimensional, depth integrated, unsteady, free-surface model based on the shallow water equations. The development was motivated by the desire of balancing computational efficiency and accuracy by selective and conjunctive use of different numerical techniques. The base framework of the discrete model uses Godunov methods on unstructured triangular grids, but the solution technique emphasizes the use of a high-resolution Riemann solver where needed, switching to a simpler and computationally more efficient upwind finite volume technique in the smooth regions of the flow. Explicit time marching is accomplished with strong stability preserving Runge-Kutta methods, with additional acceleration techniques for steady-state computations. A simplified mass-preserving algorithm is used to deal with wet/dry fronts. Application of the model is made to several benchmark cases that show the interplay of the diverse solution techniques.

Strong Stability Preserving Two-step Runge–Kutta Methods

We investigate the strong stability preserving (SSP) property of two-step Runge–Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present explicit TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order weighted essentially non-oscillatory discretizations.

Optimal implicit strong stability preserving Runge–Kutta methods

Strong stability preserving (SSP) time discretizations were developed for use with spatial discretizations of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified timestep restriction. We turn to implicit Runge–Kutta methods to alleviate this timestep restriction, and present implicit SSP Runge–Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of SSP Runge–Kutta methods) and up to eleven stages. The numerically optimal methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge–Kutta methods are diagonally implicit. These methods allow a larger SSP timestep, compared to explicit methods of the same order and number of stages. Numerical tests verify the order and the SSP property of the methods.