"advance the solution" is correct and usable in written English.
You can use it when you are suggesting or proposing a course of action that will further the progression or development of a solution. For example, "We must advance the solution by proposing new ideas and strategies."
Congress, which is just waking up to this issue, can advance the solution by putting its money and muscle behind proven programs that have been shown to re-engage the young people who have dropped out and that keep at-risk children on track to complete their educations.
To advance the solution one time step, one must solve a discrete system of nonlinear equations.
Lastly, using fluxes as boundary conditions, a dynamic two-phase flow solver is used to advance the solution in time.
To advance the solution in time, we use a MUSCL-type scheme which couples a partial volume representation with a level set one to build the numerical fluxes.
The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size.
We present a strongly-coupled solution algorithm where the high order L-stable explicit first-stage singly diagonally implicit Runge Kutta (ESDIRK) schemes are used to advance the solution in time within each separate fluid and solid subdomains.
Equally important, we can advance the solutions, like solar and wind power, that will protect our air and water and offer enduring well-paying jobs.
Continuous development and merging of analytical with semi-analytical boundary element solution methods have advanced the solution range.
The new algorithm advances the solution in the fluid domain with a standard upwind scheme and explicit time-stepping.
Solutions are obtained by a Fourier spectral element method in conjunction with a semi-implicit method combining the second-order Adams-Bashforth and first-order backward Euler schemes for advancing the solution in time.
The IMEX schemes consist of the explicit first-stage singly diagonally implicit Runge Kutta (ESDIRK) schemes, for advancing the solution in time within each separate fluid and solid subdomain, and the explicit Runge Kutta (ERK) schemes, for explicit integration of part of the coupling terms.
