You can flip your question around and ask: Why compute exact or analytic solutions?

Exact and analytic solutions are of great educational, academic, theoretical, comprehension and validation value, but their quantitative results for a given practical problem at hand are at best approximate due to an inevitable mismatch between ideal theory and observable reality.

In practice, when quantitative results for a given real-world problem are required, the numerically approximate estimation may often be demonstrably better (more accurate, faster, comprehensive, correct, exhaustive, efficient, cost-effective, etc.) than the analytical.

Numerical methods are used in those cases when finding an analytical solution is impossible. This statement is valid for chaotic systems described by systems of nonlinear ODE, for example the well-known Rossler or Lorenz systems. Obtaining an adequate model for such systems is possible only with the use of high accuracy order numerical methods.

So your point is we calculate numerical solutions to obtain adequate models, one reason is that.

This is a quite general question. We often are able to get exact analytical solutions only from simplified models that can suffer from the initial assumptions in being applied in general. This way, also analytical solutions are approximated if the global problem was simplified. Conversely, we often get numerical solutions from complex systems of equations that are less subjects to modelling hypotheses. Often analytical solutions are used as test-model for the numerical methodology one develops. Testing consistence, stability and convergence as well as the correct rate of convergence is generally well suited using well known analytical solutions.

In theory, numerical solutions are "exact" analytical solution of the so-called modified equation. Getting an accurate solution means that the modified equations can be made tending towards the original equation governing the physical phenomenon

As Prof. Denaro points out, approximate numerical solutions are obtained for problems with exact analytical solutions for the purposes of benchmarking and testing.

Only then can such numerical solutions be extended for use with more complex and realistic problems, where often no exact solutions are available.

I hope this helps!

Sometimes it might take time to find an analytical solution and it might take less time to solve an equation numerically by solving with a computer rather than a pen. One should choose the most appropriate method in order to eliminate the error. More digits lead a higher order of error by depending on method on use. Once the method is adequately chosen, the order become lower and the result become close to the exact value.

Most of system of equations consists of partial differential equations or ordinary differential equations. Such equations are mostly unsolvable by analytical methods, thus one should iterate them with numerical methods. Approximate solutions are numerical. Serial expansions must be used to solve the value of π or log values. Numerical integrals must be solved for a chaotic system.

@Sakeel A.Khan

For two reasons::

1. To check the correct form of analytical result.

2.No analytical results , then one has to rely

on analytical results.But it should be checked with limiting results.

B.Rath

We go for numerical solution if and when it is not possible to find theoretical solution. Numerical solution may be approximate or may be almost accurate. If it is possible to find theoretical solution then we usually don't go for numerical solution. However, if theoretical solution exists but difficult to find it then we usually go for numerical solution first and then for theoretical solution in order to be confident on the solution.

sometimes, computational cost involved in calculation of the exact analytical solution is too high, however, an approximate solution can be computed for the problem with much lesser number of computational steps which satisfies the accuracy threshold for an acceptable solution for the problem, in such situation an approximate solution is all that is needed and not the exact analytical solution.

for example, the condition number of a square matrix(in any norm), involves calculation of the norm of the matrix which is denoted by A, and also the norm of the inverse of A, It is well known that computing the inverse of an arbitrary matrix is computationally expensive task, but if one has to compute the exact condition number of A, this quantity has to be computed. However, most problems involving the condition number requires only an ESTIMATE of the condition number, so, by using an algorithm called Hager's Algorithm, it is possible to estimate the norm of inverse of A , without explicitly computing the Inverse of the matrix A, and thus an estimate of the condition number of the matrix A can be obtained.

Dear Shakeel Ahmed Khan,

Your question is very interesting, but the answer is not simple. For example, physical values in the macro world may be known at all points of space and time. They are then described by continual or analytic functions. Also, physical values can only be known at some points of space and time and then are described by discrete functions or numeric values. Then, mathematical equations describing physical laws may have the exact/analytic solutions, which are rarers and usually describe trivial interactions. Much more common is the case that mathematical equations do not have or can not find exact/analytic solutions and then are used approximate solutions, if it possible. Often when neither approximate solutions are satisfactory or it is impossible to find, then are found numerical solutions. Again, there is a special class of approximate solutions in some problems where the equations that describe these problems are approximate, such as various parameterizations.

For more information, see the link:

Book Numerical Method in Meteorology: Solved Problems

Best regards,

Milivoj B. Gavrilov

I agree with Debopam Ghosh.

Thanks to him for his answer with suitable example.

I agree with the interesting answer given by Milivoj B. Gavrilov also. Thanks to him for his answer with providing a link for reading material..

You can flip your question around and ask: Why compute exact or analytic solutions?

Exact and analytic solutions are of great educational, academic, theoretical, comprehension and validation value, but their quantitative results for a given practical problem at hand are at best approximate due to an inevitable mismatch between ideal theory and observable reality.

In practice, when quantitative results for a given real-world problem are required, the numerically approximate estimation may often be demonstrably better (more accurate, faster, comprehensive, correct, exhaustive, efficient, cost-effective, etc.) than the analytical.

I do agree with Ronald Kessel.

Thanks to him.

Good question and informative answers

In the field of PDE, the key when one already has an analytical solution and perform a numerical solution is due to the fact that we can evaluate the local truncation error of your numerical discretization without having the exact solution but the O(LTE) does not necessarily provide the discretization error magnitude, that is the difference between the exact and the numerical solution. Thus, a standard way to control the quality of a code is to check it against the analytical solution and compute the discretization error. If repeated over via via refined grids, this allows you to check the convergence slope.

Numerical solutions are tools for engineers to answer practical problems, analytical solutions are the job of applied mathematicians looking for general laws behind a physical phenomenon

When exact solutions are far-fetched, then approximations are a must.

Algorithms are dominated to reach the best answer within a reasonable time complexity.

Best regards

Thanks to Dhiaa Jabbar for appreciating the question and the answers given.

Thanks to Vera Maura Fernandes de Lima for recommending my answer.

Best wishes to her.

When we need quantitative answer we have to deal with computability, and you can speak of numerical solution when you can dimonstrate that it converges to the "exact" one so there ore no striking contrapposition between numerical and analitical solutions

The comment of Roberto Bernetti is correct. I agree with it.

Thanks to him.

Thank Dhritikesh Chakrabarty

Dear Roberto Bernetti,

Welcome your response.

Thanks to Luis - Felipe Velázquez for his reasonable answer.

because we can't calculate the exact solution

in real applications, the numerical solutions are more important than analytical solution or exact solution.

When obtaining an exact solution is difficult, then we would be satisfied with an approximate solution which is more important that analytical solution