/**
* Abstract class for time stepping methods. These methods assume that the
- * equation has the form: \f$ \frac{\partial y}{\partial t} = f(t,y) \f$.
+ * equation has the form: $ \frac{\partial y}{\partial t} = f(t,y) $.
*/
template <typename VECTOR>
class TimeStepping
/**
* Purely virtual function. This function is used to advance from time @p
- * t to t+ @p delta_t. @p F is a vector of functions \f$ f(t,y) \f$ that should be
+ * t to t+ @p delta_t. @p F is a vector of functions $ f(t,y) $ that should be
* integrated, the input parameters are the time t and the vector y and the
* output is value of f at this point. @p J_inverse is a vector
* functions that compute the inverse of the Jacobians associated to the
* implicit problems. The input parameters are the
- * time, \f$ \tau \f$, and a vector. The output is the value of function
+ * time, $ \tau $, and a vector. The output is the value of function
* at this point. This function returns the time at the end of the
* time step.
*/
virtual void initialize(runge_kutta_method method) = 0;
/**
* This function is used to advance from time @p
- * t to t+ @p delta_t. @p F is a vector of functions \f$ f(t,y) \f$ that should be
+ * t to t+ @p delta_t. @p F is a vector of functions $ f(t,y) $ that should be
* integrated, the input parameters are the time t and the vector y and the
* output is value of f at this point. @p J_inverse is a vector
* functions that compute the inverse of the Jacobians associated to the
* implicit problems. The input parameters are the
- * time, \f$ \tau \f$, and a vector. The output is the value of function
+ * time, $ \tau $, and a vector. The output is the value of function
* at this point. This function returns the time at the end of the
* time step. When using Runge-Kutta methods, @p F and @ J_inverse can
* only contain one element.
/**
* Purely virtual function. This function is used to advance from time @p t
- * to t+ @p delta_t. @p f is the function \f$ f(t,y) \f$ that should be
+ * to t+ @p delta_t. @p f is the function $ f(t,y) $ that should be
* integrated, the input parameters are the time t and the vector y and the
* output is value of f at this point. @p id_minus_tau_J_inverse is a function
- * that computes \f$ inv(I-\tau J)\f$ where \f$ I \f$ is the identity matrix,
- * \f$ \tau \f$ is given, and \f$ J \f$ is the Jacobian \f$ \frac{\partial
- * J}{\partial y} \f$. The input parameters are the time, \f$ \tau \f$, and
+ * that computes $ inv(I-\tau J)$ where $ I $ is the identity matrix,
+ * $ \tau $ is given, and $ J $ is the Jacobian $ \frac{\partial
+ * J}{\partial y} $. The input parameters are the time, $ \tau $, and
* a vector. The output is the value of function at this point.
* evolve_one_time_step returns the time at the end of the time step.
*/
/**
* This function is used to advance from time @p t to t+ @p delta_t. @p f
- * is the function \f$ f(t,y) \f$ that should be integrated, the input
+ * is the function $ f(t,y) $ that should be integrated, the input
* parameters are the time t and the vector y and the output is value of
* f at this point. @p id_minus_tau_J_inverse is a function that computes
- * \f$ inv(I-\tau J)\f$ where \f$ I \f$ is the identity matrix, \f$ \tau
- * \f$ is given, and \f$ J \f$ is the Jacobian \f$ \frac{\partial
- * J}{\partial y} \f$. The input parameter are the time, \f$ \tau \f$, and
+ * $ inv(I-\tau J)$ where $ I $ is the identity matrix, $ \tau
+ * $ is given, and $ J $ is the Jacobian $ \frac{\partial
+ * J}{\partial y} $. The input parameter are the time, $ \tau $, and
* a vector. The output is the value of function at this point.
* evolve_one_time_step returns the time at the end of the time step.
*/
/**
* This function is used to advance from time @p t to t+ @p delta_t. @p f
- * is the function \f$ f(t,y) \f$ that should be integrated, the input
+ * is the function $ f(t,y) $ that should be integrated, the input
* parameters are the time t and the vector y and the output is value of
* f at this point. @p id_minus_tau_J_inverse is a function that computes
- * \f$ inv(I-\tau J)\f$ where \f$ I \f$ is the identity matrix, \f$ \tau
- * \f$ is given, and \f$ J \f$ is the Jacobian \f$ \frac{\partial
- * J}{\partial y} \f$. The input parameters are the time, \f$ \tau \f$, and
+ * $ inv(I-\tau J)$ where $ I $ is the identity matrix, $ \tau
+ * $ is given, and $ J $ is the Jacobian $ \frac{\partial
+ * J}{\partial y} $. The input parameters are the time, $ \tau $, and
* a vector. The output is the value of function at this point.
* evolve_one_time_step returns the time at the end of the time step.
*/
/**
* This function is used to advance from time @p t to t+ @p delta_t. @p f
- * is the function \f$ f(t,y) \f$ that should be integrated, the input
+ * is the function $ f(t,y) $ that should be integrated, the input
* parameters are the time t and the vector y and the output is value of
* f at this point. @p id_minus_tau_J_inverse is a function that computes
- * \f$ inv(I-\tau J)\f$ where \f$ I \f$ is the identity matrix, \f$ \tau
- * \f$ is given, and \f$ J \f$ is the Jacobian \f$ \frac{\partial
- * J}{\partial y} \f$. The input parameters are the time, \f$ \tau \f$, and
+ * $ inv(I-\tau J)$ where $ I $ is the identity matrix, $ \tau
+ * $ is given, and $ J $ is the Jacobian $ \frac{\partial
+ * J}{\partial y} $. The input parameters are the time, $ \tau $, and
* a vector. The output is the value of function at this point.
* evolve_one_time_step returns the time at the end of the time step.
*/
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