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A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the Riemann solver. [2]
Sergei K. Godunov is credited with introducing the first exact Riemann solver for the Euler equations, [3] by extending the previous CIR (Courant-Isaacson-Rees) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields.
More recent research shows that an exact series solution to the Riemann problem exists, which may converge fast enough in some cases to avoid the iterative methods required in Godunov's scheme. [4]
As iterative solutions are too costly, especially in magnetohydrodynamics, some approximations have to be made. Some popular solvers are:
Philip L. Roe used the linearisation of the Jacobian, which he then solves exactly. [5]
The HLLE solver (developed by Ami Harten, Peter Lax, Bram van Leer and Einfeldt) is an approximate solution to the Riemann problem, which is only based on the integral form of the conservation laws and the largest and smallest signal velocities at the interface. [6] [7] The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper
The HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro. [8] It restores the missing rarefaction wave by using an estimation technique, such as linearisation. More advanced techniques exist, like using the Roe average velocity for the middle wave speed. These schemes are quite robust and efficient but somewhat more diffusive. [9]
These solvers were introduced by Hiroaki Nishikawa and Kitamura, [10] in order to overcome the carbuncle problems of the Roe solver and the excessive diffusion of the HLLE solver at the same time. They developed robust and accurate Riemann solvers by combining the Roe solver and the HLLE/Rusanov solvers: they show that being applied in two orthogonal directions the two Riemann solvers can be combined into a single Roe-type solver (the Roe solver with modified wave speeds). In particular, the one derived from the Roe and HLLE solvers, called Rotated-RHLL solver, is extremely robust (carbuncle-free for all possible test cases on both structured and unstructured grids) and accurate (as accurate as the Roe solver for the boundary layer calculation).
There are a variety of other solvers available, including more variants of the HLL scheme [11] and solvers based on flux-splitting via characteristic decomposition. [12]
{{cite book}}: CS1 maint: multiple names: authors list (link)The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.
The advection upstream splitting method (AUSM) is developed as a numerical inviscid flux function for solving a general system of conservation equations. It is based on the upwind concept and was motivated to provide an alternative approach to other upwind methods, such as the Godunov method, flux difference splitting methods by Roe, and Solomon and Osher, flux vector splitting methods by Van Leer, and Steger and Warming. The AUSM first recognizes that the inviscid flux consist of two physically distinct parts, i.e., convective and pressure fluxes. The former is associated with the flow (advection) speed, while the latter with the acoustic speed; or respectively classified as the linear and nonlinear fields. Currently, the convective and pressure fluxes are formulated using the eigenvalues of the flux Jacobian matrices. The method was originally proposed by Liou and Steffen for the typical compressible aerodynamic flows, and later substantially improved in to yield a more accurate and robust version. To extend its capabilities, it has been further developed in for all speed-regimes and multiphase flow. Its variants have also been proposed.
Sergei Konstantinovich Godunov was a Soviet and Russian professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirsk, Russia.
In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations.
Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDEs). They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD).
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties:
In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite volume method which solves exact, or approximate Riemann problems at each inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space and time, yet can be used as a base scheme for developing higher-order methods.
Bram van Leer is Arthur B. Modine Emeritus Professor of aerospace engineering at the University of Michigan, in Ann Arbor. He specializes in Computational fluid dynamics (CFD), fluid dynamics, and numerical analysis. His most influential work lies in CFD, a field he helped modernize from 1970 onwards. An appraisal of his early work has been given by C. Hirsch (1979)
Computational magnetohydrodynamics (CMHD) is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques employed in Computational fluid dynamics. The complexity mainly arises due to the presence of a magnetic field and its coupling with the fluid. One of the important issues is to numerically maintain the (conservation of magnetic flux) condition, from Maxwell's equations, to avoid the presence of unrealistic effects, namely magnetic monopoles, in the solutions.
A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.
Flux-corrected transport (FCT) is a conservative shock-capturing scheme for solving Euler equations and other hyperbolic equations which occur in gas dynamics, aerodynamics, and magnetohydrodynamics. It is especially useful for solving problems involving shock or contact discontinuities. An FCT algorithm consists of two stages, a transport stage and a flux-corrected anti-diffusion stage. The numerical errors introduced in the first stage are corrected in the anti-diffusion stage.
In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficult task because such flows result in sharp, discontinuous changes in flow variables such as pressure, temperature, density, and velocity across the shock.
The Roe approximate Riemann solver, devised by Phil Roe, is an approximate Riemann solver based on the Godunov scheme and involves finding an estimate for the intercell numerical flux or Godunov flux at the interface between two computational cells and , on some discretised space-time computational domain.
In computational fluid dynamics, the volume of fluid (VOF) method is a free-surface modelling technique, i.e. a numerical technique for tracking and locating the free surface. It belongs to the class of Eulerian methods which are characterized by a mesh that is either stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface. As such, VOF is an advection scheme—a numerical recipe that allows the programmer to track the shape and position of the interface, but it is not a standalone flow solving algorithm. The Navier–Stokes equations describing the motion of the flow have to be solved separately. The same applies for all other advection algorithms.
Philip L. Roe is a Professor of Aerospace Engineering at the University of Michigan in Ann Arbor. He is known for his work in the field of Computational Fluid Dynamics and Magnetohydrodynamics. Roe made fundamental contributions to the development of high-resolution schemes for hyperbolic conservation laws. He has developed approximate Riemann solver called Roe solver for compressible flows with shocks.
Randall J. LeVeque is a Professor of Applied Mathematics at University of Washington who works in many fields including numerical analysis, computational fluid dynamics, and mathematical theory of conservation laws. Among other contributions, he is lead developer of the open source software project Clawpack for solving hyperbolic partial differential equations using the finite volume method. With Zhilin Li, he has also devised a numerical technique called the immersed interface method for solving problems with elastic boundaries or surface tension.
Burton Wendroff is an American applied mathematician known for his contributions to the development of numerical methods for the solution of hyperbolic partial differential equations. The Lax–Wendroff method for the solution of hyperbolic PDE is named for Wendroff.
The finite point method (FPM) is a meshfree method for solving partial differential equations (PDEs) on scattered distributions of points. The FPM was proposed in the mid-nineties in, and with the purpose to facilitate the solution of problems involving complex geometries, free surfaces, moving boundaries and adaptive refinement. Since then, the FPM has evolved considerably, showing satisfactory accuracy and capabilities to deal with different fluid and solid mechanics problems.
