Active Flux Method Research Articles

Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics

The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

A hybrid finite element–finite volume method for conservation laws

We propose an arbitrarily high-order accurate numerical method for conservation laws that is based on a continuous approximation of the solution. The degrees of freedom are point values at cell interfaces and moments of the solution inside the cell. To lowest (3rd) order this method reduces to the Active Flux method. The update of the moments is achieved immediately by integrating the conservation law over the cell, integrating by parts and employing the continuity across cell interfaces. We propose two ways how the point values can be updated in time: either by first deriving a semi-discrete method that uses a finite-difference-type formula to approximate the spatial derivative, and integrating this method e.g. with a Runge-Kutta scheme, or by using a characteristics-based update, which is inspired by the original (fully discrete) Active Flux method. We analyze stability and accuracy of the resulting methods.

The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement

We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be applied. The exchange of data between grid patches is organised via ghost cells. The local stencil in space and time and the availability of the point values that are used for the reconstruction, leads to an efficient implementation. The resulting method is third order accurate, conservative and allows the use of subcycling in time.

A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations

We show how to combine in a natural way (i.e., without any test nor switch) the conservative and non-conservative formulations of an hyperbolic system that has a conservative form. This is inspired from two different classes of schemes: the residual distribution one (Abgrall in Commun Appl Math Comput 2(3): 341–368, 2020), and the active flux formulations (Eyman and Roe in 49th AIAA Aerospace Science Meeting, 2011; Eyman in active flux. PhD thesis, University of Michigan, 2013; Helzel et al. in J Sci Comput 80(3): 35–61, 2019; Barsukow in J Sci Comput 86(1): paper No. 3, 34, 2021; Roe in J Sci Comput 73: 1094–1114, 2017). The solution is globally continuous, and as in the active flux method, described by a combination of point values and average values. Unlike the “classical” active flux methods, the meaning of the point-wise and cell average degrees of freedom is different, and hence follow different forms of PDEs; it is a conservative version of the cell average, and a possibly non-conservative one for the points. This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem. We also develop a method to perform non-linear stability. We illustrate the behaviour on several benchmarks, some quite challenging.

On the Active Flux Scheme for Hyperbolic PDEs with Source Terms

The Active Flux scheme is a Finite Volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver: the continuous reconstruction serves as initial data for the evolution of the points values. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. This paper focuses on the conceptual extension of Active Flux to include source terms, and thus for simplicity assumes the homogeneous part of the equations to be linear. To a large part, the treatment of the source terms is independent of the choice of the homogeneous part of the system. Additionally, only systems are considered which admit characteristics (instead of characteristic cones). This is the case for scalar equations in any number of spatial dimensions and systems in one spatial dimension. Here, we succeed to extend the Active Flux method to include (possibly nonlinear) source terms while maintaining third order accuracy of the method. This requires a novel (approximate) operator for the evolution of point values and a modified update procedure of the cell average. For linear acoustics with gravity, it is shown how to achieve a well-balanced / stationarity preserving numerical method.

The Cartesian Grid Active Flux Method: Linear stability and bound preserving limiting

The primary contribution of this article is a linear stability analysis of the two-dimensional Cartesian grid Active Flux method. For the advection equation we show that stability for CFL ≤ 1 requires a more accurate flux computation than previously assumed. For the acoustic equations we confirm the expected stability for CFL≤12. Furthermore, we introduce a nonlinear limiting based on a bound preserving multi-dimensional reconstruction. Numerical results for advection and Burgers’ equation illustrate the performance of this limiting technique.

Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids

Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

A New ADER Method Inspired by the Active Flux Method

We study numerical methods that are inspired by the active flux method of Eymann and Roe and present several new results for one and two-dimensional hyperbolic problems. For one-dimensional linear problems we show that the unlimited active flux method can be interpreted as an ADER method. This interpretation motivates the construction of new third order accurate methods for nonlinear hyperbolic conservation laws. In the two-dimensional case, equivalent methods are only obtained for scalar linear problems. For two-dimensional linear systems the methods are no longer equivalent. For the two-dimensional acoustic equations we compare the accuracy of the two resulting approaches. While commonly used methods for hyperbolic problems are based on discontinuous reconstructions, the active flux method uses a continuous, piecewise quadratic reconstruction. For nonlinear problems we identify a situation in which the continuous reconstruction leads to an unstable approximation. We propose a limiting strategy which overcomes this problem. Our limited version of the active flux method uses the same local stencil as the original method.

HIGH EFFICIENCY WELDING PROCESS FOR STAINLESS STEEL MATERIALS

The high efficiency tungsten inert gas (TIG) welding process has been developed, including active flux welding process, mixed shielded welding process and double shielded welding pro- cess, to increase the weld depth/width ratio (D/W ) of conventional TIG welding method. Compared to the active flux method, mixed shielding method can make penetration deeper and the industrializa- tion can be realized easily due to the simplification in operation. Double shielded method can avoid the oxidation of tungsten electrode. The results of experiment and simulation show that the change of the Marangoni convection direction which arises from the adjustment of the oxygen content in the weld pool is one of the main factors contributing to the increase in TIG weld penetration, and the large D/W ratio can be obtained by adjusting the active element content in the liquid pool. High efficiency TIG welding process is not sensitive to welding parameters (welding speed, welding current and electrode gap) and therefore is suitable to be applied in industry easily.