FVM Interpolation and Gradient Schemes
Flux Limiters
Flux limiters are used in high resolution numerical scheme to solve governing partial differential equations. Their main purpose is to avoid the oscillations that would occur due to shocks, discontinuities, or steep gradients in the solution domain.
It has been observed that lower-order schemes are stable but dissipative in nature around the points of discontinuity/shock while higher-order schemes are unstable in nature and oscillates around points of discontinuity.
Flux limiters helps to tune the numerical scheme using flux limiter function such that resulting scheme gives higher-order accuracy at smooth region and first-order accuracy around discontiniuties/shocks.
Interpolation Scheme
Interpolation schemes are used to approximate the property values located at the boundaries of the control volumes using interpolation nodes as the correspondent properties located at the center of the adjacent control volumes.
Inorder to find function value at faces of control volume following interpolation scheme is used
- Upwind Differencing Scheme
- Central Differencing Scheme
- Quadratic Upwind Interpolation
Upwind Differencing Scheme
Approximation by its value the node upstream of \'e\'
`phi_e={(phi_P, if vecu*vecn>0),(phi_E, if vecu*vecn<0):}`
Central Differencing Scheme
Linear interpolation between nearest nodes
`phi_e=phi_Elamda_e+phi_P(1-lamda_e)`
Quadratic Upwind Interpolation
Interpolation through parabola, three points are required
`phi_e={(phi_P+g_1(phi_E-phi_P)+g_2(phi_P-phi_W), ifu_x>0),(phi_E+g_3(phi_P-phi_E)+g_4(phi_E-phi_{E E}), if u_x<0):}`
g coefficient interms of nodal coordinate