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Here is a sketch of the proposed dcore code layout.

a cartoon of the compressible equations is

Dx/Dt + Nx = 0,

where x is the model prognostic variables (u,rho,theta), Dx/Dt = x_t + u.grad x, and N is a nonlinear operator with Nx = N(x_0) + L(x-x_0) + o(x^2), where x_0 is a reference profile with u=0, and given rho, theta. We apply further approximations to L by neglecting horizontal derivatives

Then, the timestepping scheme is an iterative method for solving the implicit system

x^* = x^n - (1-\alpha) dt Nx^n, x^+ = \Phi_{\bar{u},dt}x^*, x^{n+1} = x^+ - \alpha dt Nx^{n+1},

where \Phi_{\bar{u},dt}x^* is the numerical solution of the pure advection equation

Dx/Dt = 0,

solved over one timestep, with constant advection velocity \bar{u} = u^n + \alpha(u^{n+1}-u^n), with initial condition x^*.

To solve this iteratively, we calculate iterative approximations y^k to x^{n+1}, with y^k = x^n, using the incremental formulation

A dy^{k+1} = x^+ - dt/2Ny^k - y^k, y^{k+1} = y^k + dy^{k+1},

where x^+ is obtained by

x^+ = \Phi_{\bar{u}}x^*,

where \bar{u} = u^n + \alpha(v^k - u^n), and v^k is the velocity from y^k.

The operator A is I + alphadt(L - B),

where B is the linearisation of \Phi_{\bar{u},dt}x^* about the reference profile, i.e. \Phi_{\bar{u},x^} \approx x^ + B(x^n + \alphadt(y^k - x^n)).

Hence, the steps are: (1) Compute x^*. (2) set k=0, y^k = x^n. (3) compute \bar{u} from y^k. (4) calculate x^+ using \bar{u} by calling advection schemes independently for (u,\rho,\theta) [or possibly sequentially for conservative theta advection], (5) compute the residual x^+ - dt/2Ny^k - y^k, (6) solve the linear system for dy^{k+1}, (7) update y^{k+1} = y^k + dy^{k+1}. (8) increment