And also the scheme should be first order accurate and it should not produce spurious oscillations in any case. The advection equation has positive wave speed.
There are many methods that are less dissipative than FTBS. A nice introduction to higher accuracy finite volume methods is in the classical paper
LeVeque, Randall J. "High-resolution conservative algorithms for advection in incompressible flow." SIAM Journal on Numerical Analysis 33.2 (1996): 627-665.
A more complex and advanced method is proposed in my own work
Restelli, Marco, Luca Bonaventura, and Riccardo Sacco. "A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows." Journal of Computational Physics 216.1 (2006): 195-215,
but there are literally hundreds of methods for the purpose you mention.
As Prof. Bonaventura said, there are literally hundreds of methods for the purpose you mention. And I think for basic concepts of finite volume methods this book could be useful:
Computational Fluid Dynamics: An Introduction for Engineers, M. B. Abbott and D. R. Basco
Thanks for the responses but I was actually looking for a "first order scheme" which is less dissipative than FTBS and it is mentioned in the question. I know about high resolution schemes but they switch between 2nd and 1st order accuracies depending upon the solution features to avoid oscillations. I am particularly looking for an inviscid compressible flow situation governed by hyperbolic equation (eg, Euler equations) and linear advection equation is the simplest model for such hyperbolic equation.
I want to further elaborate that in high resolution methods, for example, in Sweby's flux limited method the scheme adapts itself by varying between a first order scheme like FTBS and a second order scheme like Lax-Wendroff method and even extrapolating depending on solution gradients but satisfying a nonlinear stability condition like TVD such that spurious oscillations are bounded.
Now for the linear advection equation (linear implies constant wave speed and say a positive speed) I know that FTBS is the simplest first order accurate scheme and I am looking for an another first order accurate or close to first order accurate (but not 2nd order accurate scheme) which does not allow any spurious oscillations in any case.
Thank you for giving more details on what your target is. In this case, you may consider schemes like the one proposed in this paper
Liu, Xu-Dong. "A maximum principle satisfying modification of triangle based adapative stencils for the solution of scalar hyperbolic conservation laws." SIAM journal on numerical analysis 30.3 (1993): 701-716.
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