  We present a new class of component-wise numerical schemes that are in the
family of relaxation formulations, originally introduced by [S. Jin and Z. P.
Xin, Comm. Pure Appl. Math., 48(1995), pp. 235-277]. The relaxation framework
enables the construction of schemes that are free of nonlinear Riemann solvers
and are independent of the underlying eigenstructure of the problem. The
constant relaxation schemes proposed by Jin & Xin can however introduce strong
numerical diffusion, especially when the maximum characteristic speeds are high
compared to the average speeds in the domain. We propose a general class of
variable relaxation formulations for multidimensional systems of conservation
laws which utilizes estimates of local maximum and minimum speeds to arrive at
more accurate relaxation schemes, irrespective of the contrast in maximum and
average characteristic speeds. First and second order variable relaxation
methods are presented for general nonlinear systems in one and two spatial
dimensions, along with monotonicity and TVD (Total Variation Diminishing)
properties for the 1D schemes. The effectiveness of the schemes is demonstrated
on a test suite that includes Burgers' equation, the weakly hyperbolic
Engquist-Runborg problem, as well as the weakly hyperbolic gas injection
displacements that are governed by strong nonlinear coupling thus making them
highly sensitive to numerical diffusion. In the latter examples the second
order Jin-Xin scheme fails to capture the fronts reasonably, when both the
first and second order variable relaxed schemes produce the displacement
profiles sharply.
