Explicit Strong Stability Preserving Runge-Kutta Research Articles

A new high-order maximum-principle-preserving explicit Runge–Kutta method for the nonlocal Allen–Cahn equation

We develop the high-order maximum principle preserving schemes based on combining the scalar variable with the explicit strong stability preserving Runge–Kutta (eSSPRK) methods, which can be applied to a class of gradient flows. In this work, we use the methods to deal with the nonlocal Allen–Cahn equation. There are two remarkable properties of the scheme. On the one hand, by using the introduced scalar variable to control the nonlinear term, and adopting the eSSPRK method to temporal discretization, we only need to solve the linear equations with constant coefficients at each time level. On the other hand, we introduce the first-order approximation of the energy balance equation instead of the dynamic equation for the scalar variable, and we demonstrate that the proposed schemes are unconditionally energy stable. In particular, the discrete schemes are verified to preserve the maximum bound principle under the time-step restriction. Moreover, error estimates and the asymptotic compatibility of the proposed methods are proved, respectively. Numerical examples are performed to illustrate the efficiency and accuracy of the numerical schemes.

A High-Order Velocity-Based Discontinuous Galerkin Scheme for the Shallow Water Equations: Local Conservation, Entropy Stability, Well-Balanced Property, and Positivity Preservation

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas. In this work, we present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor (in: Tezduyar, Hughes T (eds) Proceedings of the winter annual meeting of the American Society of Mechanical Engineering 1986) back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions. The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge–Kutta (SSP–RK) time integrators (Gottlieb et al. in SIAM Rev. 43, 89–112, 2001) to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter (Cockburn and Shu in J Comput Phys 141:199–224, 1998) on the conservative variables using the Fu-Shu troubled cell indicator (Fu and Shu in J Comput Phys 347:305–327, 2017) in each inner stage of the Runge–Kutta time stepping to suppress numerical oscillations. This fully discrete can be readily applied to various SWEs simulations without dry areas where the water height is close to zero. The case with dry areas need further special attention, where the velocity approximation can be unphysically large near cells with a small water height, which may eventually crashes the simulation if no special treatment is used near these cells. Here we propose a simple wetting/drying treatment for the velocity update without violating the local conservation property to enhance the robustness of the overall scheme. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

Monolithic parabolic regularization of the MHD equations and entropy principles

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

Embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods

We construct a family of embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods of order 2≤p≤4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective property, large stability region, and small error values as defined in Dekker and Verwer (1984) and Kennedy et al. (2000). The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size. Through several numerical experiments, we assess the overall effectiveness in terms of work versus precision while also taking into consideration accuracy and stability.

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

A high-order partitioned fluid-structure interaction framework for vortex-induced vibration simulation

Vortex-induced vibration (VIV) is a highly nonlinear fluid-structure interaction (FSI) phenomenon, and its accurate prediction remains challenging. In this study, we developed a high-order partitioned FSI framework towards accurate simulation of the VIV phenomenon. The focus is on simulating the two-way coupled interaction between a flexibly mounted rigid body and its surrounding fluids. In the FSI framework, a high-order computational fluid dynamics method, i.e., flux reconstruction/correction procedure via reconstruction, is utilized to simulate the unsteady compressible Navier–Stokes equations in the arbitrary Lagrangian Eulerian format on unstructured moving grids. A linearized fluid-solid Riemann solver is employed to weakly couple fluid dynamics and rigid-body dynamics on the fluid-solid interface, which can effectively mitigate the so-called added-mass instability. Time marching is conducted through either the linearly implicit Rosenbrock–Wanner method or the explicit strong stability preserving Runge–Kutta method. The high-order FSI framework is verified with a challenging VIV phenomenon, i.e., VIV of a zero-mass cylinder in viscous flows, and has been used to study two VIV-related problems, namely, the nonlinear energy sink and oscillating foil based energy harvesting mechanism.

Locally bound-preserving enriched Galerkin methods for the linear advection equation

In this work, we introduce algebraic flux correction schemes for enriched (P1⊕P0 and Q1⊕P0) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P1/Q1 component in the admissible range, we limit the fluxes and element contributions, the complete removal of which would correspond to first-order upwinding. The first limiting procedure that we consider in this paper is based on the flux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-defined residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge–Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems.

A conservative numerical scheme for modeling nonlinear acoustic propagation in thermoviscous homogeneous media

A nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented. This system constitutes a full nonlinear wave model that has been formulated in the conservation form. Initially, this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the Lax–Wendroff theorem, and the scheme can match the solutions of the Westervelt and Burgers equations numerically. Here, high-order numerical descriptions of strongly nonlinear wave propagations become of great interest. For that matter we consider finite difference formulations of the weighted essentially non-oscillatory (WENO) schemes associated with explicit strong stability preserving Runge–Kutta (SSP-RK) time integration methods. Although this strategy is known to be computationally demanding, it is found to be effective when implemented to be solved in graphical processing units (GPUs). As we consider wave propagations in unbounded domains, perfectly matching layers (PML) have been also considered in this work. The proposed system model is validated and illustrated by using one- and two-dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media.

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.