Upwind scheme

In computational physics, upwind schemes denote a class of numerical discretization methods for solving hyperbolic partial differential equations. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate the direction of propagation of information in a flow field. The upwind schemes attempt to discretize hyperbolic partial differential equations by using differencing biased in the direction determined by the sign of the characteristic speeds. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Rees who proposed the CIR method. In computational physics, upwind schemes denote a class of numerical discretization methods for solving hyperbolic partial differential equations. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate the direction of propagation of information in a flow field. The upwind schemes attempt to discretize hyperbolic partial differential equations by using differencing biased in the direction determined by the sign of the characteristic speeds. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Rees who proposed the CIR method. To illustrate the method, consider the following one-dimensional linear advection equation which describes a wave propagating along the x {displaystyle x} -axis with a velocity a {displaystyle a} . This equationis also a mathematical model for one-dimensional linear advection. Consider a typical grid point i {displaystyle i} in thedomain. In a one-dimensional domain, there are only two directions associated with point i {displaystyle i} – left (towards negative infinity) andright (towards positive infinity). If a {displaystyle a} is positive, the travelling wave solution of the equation above propagates towards the right, the left side of i {displaystyle i} is called upwind side and the right side is the downwind side. Similarly, if a {displaystyle a} is negative the travelling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivative, ∂ u / ∂ x {displaystyle partial u/partial x} contains morepoints in the upwind side, the scheme is called an upwind-biased or simply an upwind scheme. The simplest upwind scheme possible is the first-order upwind scheme. It is given by