Task: Question Answering
Input (English): Context: In aeronautics, transonic refers to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.0, i.e. at sea level. This condition depends not only on the travel speed of the craft, but also on the temperature of the airflow in the vehicle's local environment. It is formally defined as the range of speeds between the critical Mach number, when some parts of the airflow over an air vehicle or airfoil are supersonic, and a higher speed, typically near Mach 1.2, when most of the airflow is supersonic. Between these speeds some of the airflow is supersonic, but a significant fraction is not., A Riemann solver is a numerical method used to solve a Riemann problem . They are heavily used in computational fluid dynamics and computational magnetohydrodynamics ., A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any given point is specified by both a "direction" and a "magnitude" (or strength); as such it is a vector field. The term is used for two distinct but closely related fields denoted by the symbols and , where is measured in units of amperes per meter (symbol: Am or A/m) in the SI. is measured in teslas (symbol: T) and newtons per meter per ampere (symbol: NmA or N/(mA)) in the SI. is most commonly defined in terms of the Lorentz force it exerts on moving electric charges., In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial function is a function that is a polynomial on each of its sub-domains, but possibly a different one on each., Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio technologies such as power generation, electric motors, wireless communication, cameras, televisions, computers etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents and changes of each other. One important consequence of the equations is that fluctuating electric and magnetic fields can propagate at the speed of light. This electromagnetic radiation manifests itself in manifold ways from radio waves to light and X- or -rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations, and first proposed that light is an electromagnetic phenomenon., Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial experimental validation of such software is performed using a wind tunnel with the final validation coming in full-scale testing, e.g. flight tests., A Riemann problem, named after Bernhard Riemann, consists of an initial value problem composed of a conservation equation together with piecewise constant data having a single discontinuity. 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., Georg Friedrich Bernhard Riemann (17 September 1826  20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.
His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded, although it is his only paper in the field, as one of the most influential papers in analytic number theory.
Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity., Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics)., Magnetohydrodynamics (MHD; also magneto fluid dynamics or hydromagnetics) is the study of the magnetic properties of electrically conducting fluids. Examples of such magneto-fluids include plasmas, liquid metals, and salt water or electrolytes. The word "magnetohydrodynamics (MHD)" is derived from "magneto-" meaning magnetic field, "hydro-" meaning water, and "-dynamics" meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970., A supercomputer is a computer with a high level of computational capacity compared to a general-purpose computer. Performance of a supercomputer is measured in floating-point operations per second (FLOPS) instead of million instructions per second (MIPS). As of 2015, there are supercomputers which can perform up to quadrillions of FLOPS., 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 formula_1 (conservation of magnetic flux) condition, from Maxwell's equations, to avoid any unphysical effects., In mathematics, in the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time., A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. A wind tunnel consists of a tubular passage with the object under test mounted in the middle. Air is made to move past the object by a powerful fan system or other means. The test object, often called a wind tunnel model, is instrumented with suitable sensors to measure aerodynamic forces, pressure distribution, or other aerodynamic-related characteristics., Subject: riemann solver, Relation: named_after, Options: (A) 8 (B) air (C) bernhard riemann (D) geometry (E) james clerk maxwell (F) main (G) mathematics (H) point (I) salt (J) speed of sound (K) up (L) water (M) wind
Output (English): bernhard riemann