diff --git a/doc/sphinx/reference/onedim/nonlinear-solver.md b/doc/sphinx/reference/onedim/nonlinear-solver.md index 66bb8e03aa..d4fef8e580 100644 --- a/doc/sphinx/reference/onedim/nonlinear-solver.md +++ b/doc/sphinx/reference/onedim/nonlinear-solver.md @@ -65,7 +65,7 @@ $$ J(x) =\pxpy{F(x)}{x} $$ -For the vector-value residual $F(x)$, the Jacobian matrix has elmenets that are +For the vector-value residual $F(x)$, the Jacobian matrix has elements that are partial derivatives of the residuals with respect to each solution component at each grid point. @@ -160,7 +160,7 @@ $$ During the search for the correct value of $\lambda$, the value of $\lambda$ starts at 1, it is adjusted down to a value that keeps the solution within the trust region. The process then begins for finding $\lambda$, failures result in the damping factor -being reduced by a constant factor. The current factor in Cantera is the $\sqrt{2}$. +being reduced by a constant factor. The current factor in Cantera is the $\sqrt{2}$. During the damped Newton method, the Jacobian is kept at the $x^{(k)}$ value. This sometimes can cause issues with convergence if the Jacobian becomes out of date @@ -284,7 +284,7 @@ F(x_n, x_{n+1}) = -\frac{\alpha}{\Delta t}(x_{n+1} - x_n) + F_{ss}(x_{n+1}) $$ For the Newton method, we linearize the residual equation about the solution vector at -the next iteration(not timestep) by using a Taylor series expansion. The linearized +the next iteration (not time step) by using a Taylor series expansion. The linearized equation is given by: $$ @@ -339,8 +339,8 @@ J_{ss}(x^{(k)}) \Delta x^{(k)} = -\lambda^{(k)} F_{ss}(x^{(k)}) $$ The transient equation has the same form as the steady-state equation, and so the same -damped Newton method can be used to solve the transient problem for a single timestep. +damped Newton method can be used to solve the transient problem for a single time step. $$ J(x_{n+1}^{(k)}) \Delta x_{n+1}^{(k)} = -\lambda^{(k)} F(x_n, x_{n+1}^{(k)}) -$$ \ No newline at end of file +$$