I do not know the meaning of your word "best". However, good and fruitful algorithms can be developed on the existing tools as follows. For ODE, you may apply RK-IV; for PDE, Finite Element Method (FEM); for complex polynomial equations, Decomposition Method (DM); and so on.

Some suggested readings are:

1. J.C. Tannehill et al., COMPUTATIONAL FLUID MECHANICS AND HEAT TRANSFER (Taylor & Francis, USA, 1994).

2. Lindfield, G.R., Penny, J.E.T.: NUMERICAL METHODS USING MATLAB. Elsevier, Amsterdam (2012).

For ODE, it is better to use high order methods with variable step, for example, Dormand-Prince method 8, or extrapolation algorithms.

this depends on the problem to be studied,type of loading,material,etc.

Most nonlinear solution algorithms are best suited for a certain class of problems.

Newtons method: if the hessian is easy to compute and evaluate

Quasi Newton Methods e.g Gauss Newton Algorithm: for cases where you do not have an hessian, you pay with convergence

conjugate gradient methods: good for large sparse matrices

Trust region methods: better for quadratic like problems

For ODES

this depends on the order of error you're willing to tolerate

ODE45, Dormand Prince

or if youre dealing with a stiff differential equation then you need ODE23s or ODE15s in MATLAB.