The most important technology in FVM is to reconstruct the variables at cell interfaces accurately. So why not directly solve PDE on cell interfaces to get accurate value and their evolution? Do you believe that it will be much accurate than the reconstruction method? We can directly solve PDE not only on the cell, but also on the cell interfaces. For example, in the UGKS presented by Xu and Huang and DUGKS presented by Guo et al., they solved the Boltzmann equation on cell interfaces.
The difficuty may come from the stencil problem.
The stencil on cell interface is so compact comparing with cell center. But if we using the centered type DG, I think this will be not a problem again.
In fact, there is a type of Structure Preserved method which shares this idea.
In the future, I believe that this idea will be accepted by more and more researchers. Specially, for the sigularity problems.
your opinion some how resembles the Godunov schem. that is a very powerful method, but while you have discreted domain, there will be a first order error. higher order extensions are also availabel, such as MUSCL method. but sometimes calculating the exact solution for each cell will need more time than reconstruction and approximate flux functions.
Godnuov scheme is just roughly approximation solution. What I want is the accurate solution of NS equation without any scheme. Directly solve the PDE on cell center and cell interfaces. MUSCL is just a kind of technology to get high order initial state values at the left and right hand sides around the cell interfaces, which does not connect with my question here.
first, Godunov scheme is not a "roughly approximate solution", and it can be considered as the most accurate and robust first order scheme in some cases (e.g.,shock capturing). the reason i mentioned the MUSCL was to name one higher order extension of the Godunov scheme, MUSCL can use exact Riemann solvers in the case of Euler equations.
you did not mention that you mean NS equations, i think the reason these equations cant be treated the way you want, is simply that there is No exact solution for one cell. that's why non-linear PDEs are to be solved approximately by numerical techniques.
by the way, by using Godunov-like Schemes, you can use the exact solution of inviscid terms, then adding the viscous fluxes.
If you use Godunov-like Schemes, of course, you can get inviscid terms, but it is not the exact one, since this is no general multi-dimensional Godunov-like Schemes. All these schemes (Riemann solvers) are only approximate! You should note this! Moreover, when we calculate the convection and diffussion fluxes by two steps, we implicitly assumes that this two parts can be seperated, which in fact, is some kind of splitting method. Even you use accurate Euler solutions (multidimensional), it should be some error inside. Since when the accurate Euler solutions is used to obtaine the inviscous fluxes, there have an implicit constrain. Namely, the variables satisfy the Euler equation. but in fact, the variables on the cell interface should be constrained by the NS equation. That's why I ask this questions.
not all the Riemann solvers are approximate, approximate Riemann solvers are a group [the book Riemann solvers and ... by E.F.Toro], i agree with other statements in your last answer, but i told you there is no exact solution for the NS equations, if there were any, of course there was a Godunov type method to solve the NS equations without splitting.
Please give me the exact Riemann solver. Even for the GRP and the ADER， they are still appoximate，but more accuract than others. There is no analytic solution of NS， but we can solve the NS equation numerically on cell interfaces. This is my idea.
Godunov is credited with the first exact Riemann solver for the Euler equations. Chorin, van Leer, Smoller, Dutt, Gottlieb and Groth are some other people who presented exact Riemann solvers [Riemann solvers and computational methods for fluid dynamics, E.F Toro, third edition, p116].
by the word "GRP" if you mean "Generalized Riemann problem", i cant understand it's relation with exact Riemann solvers, and also ADER is a method to solve the Euler equations using the GRP.
Riemann solver is a solution to the shock tube problem, which there exists many approaches to find the exact solution both numerically and analytically.
What you are talking about is only 1D, but I hope to discuss multi-dimensional problems.
CE/SE is not solving the equations on cell interface. It use staggered mesh and use Taylor expansion with respect to time and space to obtain the variables on cell interfaces and then use the Cauchy-Kowalewski procedure to transfer the time related derivatives into the space derivatives. The CE/SE is compact and low dissipation. But it is still not satisfying.
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