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What is the importance of discontinuous Galerkin method for a nonconservative form of hyperbolic PDE?

a)What are the advantages?

b)what are the main differences when we are using finite volume and Galerkin method for a nonconservative form of hyperbolic PDE?

How to find out the wave speed of u(x,t)? , where x is the space variable, t is the time variable, and u is the solution wave of corresponding hyperbolic PDE.

Numerical solution to PDEs through different schemes (implicit or explicit) results in some error which are referred as dissipative or dispersive error for the solution of Hyperbolic or Parabolic PDE. How these are verified and minimised for more accurate results

I'm looking for an algorithm for solving (and representation of non-zero elements) linear systems with large sparse unsymmetric matrices (not diagonally dominant).

I'm interested in ONLY DIRECT method.

(The matrices are obtained from FDM for hyperbolic PDE).

In general, air flow can be classified three kinds of method at the partial differential equation(PDE), and they are elliptic, parabolic, hyperbolic.

Parabolic is utlized for one-way flow such as wave propagation toward open domain (M=1). 

Elliptic is utilized for commuicational flow, affect each other for all of the points, such as wave propagation in subsonic (M<1).

In case of boundary layer, marching to downstream until before separation, it is applied parabolic way in general.

I have a question at this point. Boundary layer(BL) almost occured under incompressible flow(M<0.3) and always under M<1. However BL theory almost be applied parabolic property, not elliptic despite of being under subsonic flow.

Commonsensically,  based on the charater of BL, it is fair but theoretically hard to be understood to me.

DFN Approach for Fractured Reservoirs: Suitable at local-scale? DFN approach (a) may easily incorporate field-data; (b) may render near-realistic fracture networks that may possibly include hydraulically connected fractures, I suppose (because, two-dimensional fracture network from a top view, covering the areal extent will not be any use; and what we require is the actual three-dimensional fracture network with dip details, which are essentially hydraulically connected); and (c) may explicitly represent the fracture attributes (but still, getting the actual details on the distributions of fracture length, fracture width, fracture spacing, fracture aperture thickness, fracture dip remain extremely challenging); and in turn DFN approach may work well for a larger field-scale scenario, where the local-scale interaction between low-permeable rock-matrix and high-permeable fracture remains not required. However, in petroleum reservoirs, all the hydrocarbon has been stored within the low-permeable rock-matrix; and hence, the interaction between local-scale rock-matrix and fracture becomes very crucial in order to expel the oil out of the rock-matrix; and subsequently,   ensuring the continuity of fluid mass fluxes at the fracture-matrix interface becomes very challenging using any numerical approach at the local-scale. This transient interaction between fracture and matrix remains very crucial until a hydrodynamic equilibrium is achieved between fracture and matrix. This precision of deducing an accurate fluid mass flux at the fracture-matrix interface that subsequently ensures the fluid mass continuity between fracture and matrix – is what exactly required – and this will help to precisely estimate the RF associated with a fractured reservoir. So, what do we need to get an improved/precise RF from a Fractured Reservoir? (a) Reservoir Geo-mechanics (stress and strain - displacement of fracture surfaces with details on zero aperture thickness) (b) Fluid flow (pressure / saturation) (c) Role of differential advection and effective diffusion (d) Scale-dependent heterogeneity (Deducing a reasonable REV) (e) Nature of hyperbolic dominant PDE (f) Nature of hyperbolic dominant source term and, we require much more...........before we can include our investigation on the role of gravity-drainage, imbibition and capillarity. Of course, we can always approximate the above problem using Equivalent Continuum rather than attacking the problem with multi-continuum. Even with multi-continuum, we can always approximate using dual-porosity rather than using dual-permeability (if the rock-matrix is no more permeable, then, how long will it take for the oil/gas to get expelled out of rock-matrix using concentration gradient?). Even with dual-porosity, we can always assume a reasonable shape factor that reflects the fluid mass transfer between fracture and matrix; rather than going for actual 'rate limited fluid mass transfer'.

Blood Flow:

Do we have a proper momentum conservation equation for characterizing blood flow?

1.  Whether the dimensionless Womersley (Wo) number can be used to describe the transient nature of blood flow in response to a transient local pressure gradient - when the internal cross section of blood flow remains to be different from circular cylinder?

For varying cross sections, when Wo remains to be less than unity, whether the velocity profile still exhibits the parabolic shape such that the fluid oscillating with the greatest amplitude remains to be farthest from the artery walls?

Similarly, when Wo remains to be greater than unity (where, the velocity profiles are no longer parabolic), how exactly the blood remains to be phase-shifted in time relative to the oscillating pressure gradient?

If blood flow is characterized by hyperbolic PDEs (rather than parabolic PDEs), then, why not the pressure pulse of our body is no more conserved throughout our life time? Why does it follow a typical parabolic PDE pattern, where the pressure pulse keeps decaying with time, and finally, reaching a steady-state (last breathe, where the pulse becomes flattened) upon reaching a larger time level?

2.  What is the scale at which we got to look at the problem of blood flow in our body?

Will it be microscopic-scale; or, macroscopic-scale?

If so, would it remain feasible to deduce a representative blood concentration over a definite REV (Representative Elementary Volume)?

For that matter, can we deduce a reliable REV in a blood circulation system?

If not, how could we apply the conventional PDEs that remain applicable for a function that is supposed to remain continuous and smooth?

If blood flow has its importance over various scales, then, how could we characterize the blood flow using a single-continuum?

If multiple-continuum needs to be followed, then, how many continuum would be required to characterize the blood flow?

If multiple-continuum exists, then, how could we ensure the continuity of blood fluxes at the interfaces between any two continua?

Feasible to deduce proper boundary conditions for blood flow through arteries, capillaries and veins?

If both Navier-Stokes equation (momentum conservation to characterize flow through pipes) and Darcy’s equation (momentum conservation to characterize flow through a porous medium) cannot be applied to characterize this non-Newtonian blood flow; then, is there a new momentum conservation equation used to characterize blood flow through human body?

3.   Survival after a diagnosis of HF hangs around less than 7% now. Why is it so?

4.   20 or 30 years back, HFrEF – the incidence of heart failure with reduced ejection fraction (primary myocardial injury) was dominant. However, now, HFrEF – the incidence of preserved ejection fraction (induced by comorbidities) has become dominant. Complex origin of heart failure resulting from structural, mechanical or electrical dysfunction of the heart?